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\centerline{\bigrm Dynamical Theories}
\bigskip

\centerline {\bigrm of}
\bigskip

\centerline{\bigrm Brownian Motion}
\bigskip\bigskip\bigskip

\centerline{\it second edition}
\bigskip\bigskip\bigskip

\centerline{\bigrm by}
\bigskip

\centerline{\bigrm Edward Nelson}
\bigskip

\centerline{\rm Department of Mathematics}
\medskip

\centerline{\rm Princeton University}

\newpage
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~ \\[25pt]
\noindent Copyright \copyright\ 1967, by Princeton University Press.

\noindent All rights reserved.

\bigskip 

\noindent Second edition, August 2001. Posted on the Web at

\noindent {\tt http://www.math.princeton.edu/$\scriptstyle\sim$nelson/books.html}

\newpage
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~ \\[25pt]
\centerline{\bigrm Preface to the Second Edition}
\bigskip\bigskip

On July 2, 2001, I received an email from Jun Suzuki, a recent graduate
in theoretical physics from the University of Tokyo. It contained a 
request to reprint ``Dynamical Theories of Brownian Motion'', which was
first published by Princeton University Press in 1967 and was now out
of print. Then came the extraordinary statement: ``In our seminar, we
found misprints in the book and I typed the book as a TeX file with modifications.''
One does not receive such messages often in one's lifetime.

So, it is thanks to Mr.~Suzuki that this edition appears. I modified his file,
taking the opportunity to correct my youthful English and make minor changes
in notation. But there are no substantive changes from the first edition.

My hearty thanks also go to Princeton University Press for
permission to post this volume on the Web. Together with all
mathematics books in the {\it Annals Studies} and {\it Mathematical Notes}
series, it will also be republished in book form by the Press.
\bigskip

\noindent Fine Hall

\noindent August 25, 2001

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~ \\[25pt]

\centerline{\bigrm Contents}
\bigskip

{\openup1\jot

\noindent \phantom01.\quad Apology\dotfill\quad\phantom{00}1

\noindent \phantom02.\quad Robert Brown\dotfill\quad\phantom{00}5

\noindent \phantom03.\quad The period before Einstein\dotfill\quad\phantom{00}9

\noindent \phantom04.\quad Albert Einstein\dotfill\quad\phantom{0}13

\noindent \phantom05.\quad Derivation of the Wiener process\dotfill\quad\phantom{0}19

\noindent \phantom06.\quad Gaussian processes\dotfill\quad\phantom{0}27

\noindent \phantom07.\quad The Wiener integral\dotfill\quad\phantom{0}31

\noindent \phantom08.\quad A class of stochastic differential equations\dotfill\quad\phantom{0}37

\noindent \phantom09.\quad The Ornstein-Uhlenbeck theory of Brownian motion\dotfill\quad\phantom{0}45

\noindent 10.\quad Brownian motion in a force field\dotfill\quad\phantom{0}53

\noindent 11.\quad Kinematics of stochastic motion\dotfill\quad\phantom{0}65

\noindent 12.\quad Dynamics of stochastic motion\dotfill\quad\phantom{0}83

\noindent 13.\quad Kinematics of Markovian motion\dotfill\quad\phantom{0}85

\noindent 14.\quad Remarks on quantum mechanics\dotfill\quad\phantom{0}89

\noindent 15.\quad Brownian motion in the aether\dotfill\quad105

\noindent 16.\quad Comparison with quantum mechanics\dotfill\quad111

}

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\fancyhead[CO]{APOLOGY}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Apology}

\hspace{6mm}It is customary in Fine Hall to lecture on mathematics, and
any major deviation from that custom requires a defense.

It is my intention in these lectures to focus on Brownian
motion as a natural phenomenon. I will review the theories put
forward to account for it by Einstein, Smoluchowski, Langevin, Ornstein,
Uhlenbeck, and others. It will be my conjecture that a certain portion
of current physical theory, while mathematically consistent, is
physically wrong, and I will propose an alternative theory.

Clearly, the chances of this conjecture being correct are
exceedingly small, and since the contention is not a mathematical
one, what is the justification for spending time on it? The presence
of some physicists in the audience is irrelevant. Physicists
lost interest in the phenomenon of Brownian motion about thirty or
forty years ago. If a modern physicist is interested in Brownian
motion, it is because the mathematical theory of Brownian motion
has proved useful as a tool in the study of some models of quantum
field theory and in quantum statistical mechanics. I believe that
this approach has exciting possibilities, but I will not deal with
it in this course (though some of the mathematical techniques that
will be developed are relevant to these problems).

The only legitimate justification is a mathematical one.
Now ``applied mathematics'' contributes nothing to mathematics. On
the other hand, the sciences and technology do make vital contribution
to mathematics. The ideas in analysis that had their
origin in physics are so numerous and so central that analysis would
be unrecognizable without them.

A few years ago topology was in the doldrums, and then it
was revitalized by the introduction of differential structures. A
significant role in this process is being played by the qualitative
theory of ordinary differential equations, a subject having its
roots in science and technology. There was opposition on the part
of some topologists to this process, due to the loss of generality
and the impurity of methods.

It seems to me that the theory of stochastic processes is
in the doldrums today. It is in the doldrums for the same reason,
and the remedy is the same. We need to introduce differential
structures and accept the corresponding loss of generality and impurity
of methods. I hope that a study of dynamical theories of
Brownian motion can help in this process.

Professor Rebhun has very kindly prepared a demonstration
of Brownian motion in Moffet Laboratory. This is a live telecast
from a microscope. It consists of carmine particles in acetone,
which has lower viscosity than water. The smaller particles have
a diameter of about two microns (a micron is one thousandth of a
millimeter). Notice that they are more active than the larger
particles. The other sample consists of carmine particles in water---they
are considerably less active. According to theory, nearby
particles are supposed to move independently of each other, and
this appears to be the case.

Perhaps the most striking aspect of actual Brownian motion
is the apparent tendency of the particles to dance about without
going anywhere. Does this accord with theory, and how can it be
formulated?

One nineteenth century worker in the field wrote that although
the terms ``titubation'' and ``pedesis'' were in use, he preferred
``Brownian movements'' since everyone at once knew what was
meant. (I looked up these words [1]. Titubation is defined as the 
``act of titubating; specif., a peculiar staggering gait observed in
cerebellar and other nervous disturbance''. The definition of
pedesis reads, in its entirety, ``Brownian movement''.) Unfortunately,
this is no longer true, and semantical confusion can result. I
shall use ``Brownian motion'' to mean the natural phenomenon. The
common mathematical model of it will be called (with ample historical
justification) the ``Wiener process''.

I plan to waste your time by considering the history of
nineteenth century work on Brownian motion in unnecessary detail.
We will pick up a few facts worth remembering when the mathematical
theories are discussed later, but only a few. Studying the development
of a topic in science can be instructive. One realizes what
an essentially comic activity scientific investigation is (good as
well as bad).
\vfill\eject

\begin{center}
\textrm{\bf Reference}
\end{center}
[1]. Webster's New International Dictionary, Second Edition, G. 
\& C. Merriam Co., Springfield, Mass.\ (1961).

\newpage
\thispagestyle{empty}
~

%\newpage

\fancyhead[CO]{ROBERT BROWN}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Robert Brown}

\hspace{6mm}Robert Brown sailed in 1801 to study the plant life of the
coast of Australia. This was only a few years after a botanical
expedition to Tahiti aboard the \emph{Bounty} ran into unexpected
difficulties. 
Brown returned to England in 1805, however, and became a
distinguished botanist. Although Brown is remembered by mathematicians
only as the discoverer of Brownian motion, his biography
in the \emph{Encyclopaedia Britannica} makes no mention of this discovery.

Brown did not discover Brownian motion. After all, practically
anyone looking at water through a microscope is apt to see
little things moving around. Brown himself mentions one precursor
in his 1828 paper [2] and ten more in his 1829 paper
[3], starting
at the beginning with Leeuwenhoek (1632--1723), including Buffon and
Spallanzani (the two protagonists in the eighteenth century debate
on spontaneous generation), and one man (Bywater, who published in
1819) who reached the conclusion (in Brown's words) that ``not only
organic tissues, but also inorganic substances, consist of what he
calls animated or irritable particles.''

The first dynamical theory of Brownian motion was that the
particles were alive. The problem was in part observational, to
decide whether a particle is an organism, but the vitalist bugaboo
was mixed up in it. Writing as late as 1917, D'Arcy Thompson [4]
observes: ``We cannot, indeed, without the most careful scrutiny,
decide whether the movements of our minutest organisms are intrinsically
`vital' (in the sense of being beyond a physical mechanism,
or working model) or not.'' Thompson describes some motions of
minute organisms, which had been ascribed to their own activity,
but which he says can be explained in terms of the physical picture
of Brownian motion as due to molecular bombardment. On the other
hand, Thompson describes an experiment by Karl Przibram, who
observed the position of a unicellular organism at fixed intervals.
The organism was much too active, for a body of its size, for its
motion to be attributed to molecular bombardment, but Przibram concluded
that, with a suitable choice of diffusion coefficient,
Einstein's law applied!

Although vitalism is dead, Brownian motion continues to be
of interest to biologists. Some of you heard Professor Rebhun
describe the problem of disentangling the Brownian component of
some unexplained particle motions in living cells.

Some credit Brown with showing that the Brownian motion is
not vital in origin; others appear to dismiss him as a vitalist.
It is of interest to follow Brown's own account [2] of his
work. 
It is one of those rare papers in which a scientist gives a lucid
step-by-step account of his discovery and reasoning.

Brown was studying the fertilization process in a species
of flower which, I believe likely, was discovered on the Lewis and
Clark expedition. Looking at the pollen in water through a microscope,
he observed small particles in ``rapid oscillatory motion.''
He then examined pollen of other species, with similar results.
His first hypothesis was that Brownian motion was not only vital
but peculiar to the male sexual cells of plants. (This we know is
not true---the carmine particles that we saw were derived from the
dried bodies of female insects that grow on cactus plants in Mexico
and Central America.) Brown describes how this view was modified:

``In this stage of the investigation having found, as I
believed, a peculiar character in the motions of the particles of
pollen in water, it occurred to me to appeal to this peculiarity as
a test in certain Cryptogamous plants, namely Mosses, and the genus
\emph{Equisetum}, in which the existence of sexual organs had not been
universally admitted. \dots But I at the same time observed, that on
bruising the ovules or seeds of \emph{Equisetum}, which at first happened
accidentally, I so greatly increased the number of moving particles,
that the source of the added quantity could not be doubted. I
found also that on bruising first the floral leaves of Mosses, and
then all other parts of those plants, that I readily obtained similar
particles, not in equal quantity indeed, but equally in motion.
My supposed test of the male organ was therefore necessarily
abandoned.

``Reflecting on all the facts with which I had now become
acquainted, I was disposed to believe that the minute spherical
particles or Molecules of apparently uniform size, \dots
were in reality
the supposed constituent or elementary molecules of organic bodies,
first so considered by Buffon and Needham \dots''

He examined many organic substances, finding the motion,
and then looked at mineralized vegetable remains: ``With this view a minute 
portion of silicified wood, which exhibited the 
structure of Coniferae, was bruised, and spherical particles, or molecules in all
respects like those so frequently mentioned, were readily obtained
from it; in such quantity, however, that the whole substance of the
petrifaction seemed to be formed of them. From hence I inferred
that these molecules were not limited to organic bodies, nor even
to their products.''

He tested this inference on glass and minerals: ``Rocks of all ages, 
including those in which organic remains have never been
found, yielded the molecules in abundance. Their existence was
ascertained in each of the constituent minerals of granite, a fragment
of the Sphinx being one of the specimens observed.''

Brown's work aroused widespread interest. We quote from a
report [5] published in 1830 of work of Muncke in Heidelberg:

``This motion certainly bears some resemblance to that
observed in infusory animals, but the latter show more of a voluntary
action. The idea of vitality is quite out of the question.
On the contrary, the motions may be viewed as of a mechanical nature,
caused by the unequal temperature of the strongly illuminated water,
its evaporation, currents of air, and heated currents, \&c.\ ''

Of the causes of Brownian motion, Brown [3] writes:

``I have formerly stated my belief that these motions of the
particles neither arose from currents in fluid containing them,
nor depended on that intestine motion which may be supposed to
accompany its evaporation.

``These causes of motion, however, either singly or combined
with other,---as, the attractions and repulsions among the particles
themselves, their unstable equilibrium in the fluid in which they
are suspended, their hygrometrical or capillary action, and in some
cases the disengagement of volatile matter, or of minute air
bubbles,---have been considered by several writers as sufficiently
accounting for the appearance.''

He refutes most of these explanations by describing an
experiment in which a drop of water of microscopic size immersed
in oil, and containing as few as one particle, exhibits the motion
unabated.

Brown denies having stated that the particles are animated.
His theory, which he is careful never to state as a conclusion, is
that matter is composed of small particles, which he calls active
molecules, which exhibit a rapid, irregular motion having its
origin in the particles themselves and not in the surrounding fluid.

His contribution was to establish Brownian motion as an
important phenomenon, to demonstrate clearly its presence in inorganic
as well as organic matter, and to refute by experiment facile
mechanical explanations of the phenomenon.

%~ \\[1.5cm]
\begin{center}
\textrm{\bf References}
\end{center}
{\parskip=\smallskipamount

\noindent [2]. Robert Brown, \emph{A brief Account of Microscopical Observations
made in the Months of June, July, and August, 1827, on the Particles
contained in the Pollen of Plants; and on the general Existence of
active Molecules in Organic and Inorganic Bodies}, Philosophical
Magazine N. S. \textrm{4} (1828), 161--173. 

\noindent [3]. Robert Brown, \emph{Additional Remarks on Active Molecules}, Philosophical
Magazine N. S. \textrm{6} (1829), 161--166.

\noindent [4]. D'Arcy W. Thompson, ``Growth and Form'', Cambridge University
Press (1917).

\noindent [5]. Intelligence and Miscellaneous Articles: \emph{Brown's Microscopical
Observations on the Particles of Bodies}, Philosophical Magazine N. S. 
\textrm{8} (1830), 296.

}

\newpage

\fancyhead[CO]{THE PERIOD BEFORE EINSTEIN}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{The period before Einstein}

\hspace{6mm}I have found no reference to a publication on Brownian motion between 
1831 and 1857. Reading papers published in the sixties and seventies, however, one 
has the feeling that awareness of the phenomenon remained widespread (it could 
hardly have failed to, as it was something of a nuisance to microscopists). 
Knowledge of Brown's work reached literary circles. In George Eliot's ``Middlemarch'' 
(Book II, Chapter V, published in 1872) a visitor to the vicar is interested in obtaining 
one of the vicar's biological specimens and proposes a barter: ``I have some sea mice. \dots 
And I will throw in Robert Brown's new thing,---`Microscopic Observations 
on the Pollen of Plants,'---if you don't happen to have it already.''

From the 1860s on, many scientists worked on the phenomenon. 
Most of the hypotheses that were advanced could have been ruled out by consideration 
of Brown's experiment of the microscopic water drop enclosed in oil. The first to express 
a notion close to the modern theory of Brownian motion was Wiener in 1863. A little later Carbonelle claimed that the internal movements that constitute the heat content of 
fluids is well able to account for the facts. A passage emphasizing the probabilistic aspects is quoted by Perrin [6, p.\ 4]:

``In the case of a surface having a certain area, the molecular collisions of the liquid 
which cause the pressure, would not produce any perturbation of the suspended particles, 
because these, as a whole, urge the particles equally in all directions. But if the surface 
is of area less than is necessary to ensure the compensation of irregularities, there is 
no longer any ground for considering the mean pressure; the inequal pressures, continually 
varying from place to place, must be recognized, as the law of large numbers no longer 
leads to uniformity; and the resultant will not now be zero but will change continually in 
intensity and direction. Further, the inequalities will become more and more apparent 
the smaller the body is supposed to be, and in consequence the oscillations will at the 
same time become more and more brisk \dots''

There was no unanimity in this view. Jevons maintained that pedesis 
was electrical in origin. Ord, who attributed Brownian motion largely to ``the 
intestine vibration of colloids'', attacked Jevons' views [7], and I cannot refrain 
from quoting him:

``I may say that before the publication of Dr. Jevons' observations I had made many experiments to test the influence of acids [upon Brownian movements], and that my 
conclusions entirely agree with his. In stating this, I have no intention of derogating 
from the originality of of Professor Jevons, but simply of adding my testimony to his 
on a matter of some importance. \dots

``The influence of solutions of soap upon Brownian movements, as set forth by 
Professor Jevons, appears to me to support my contention in the way of agreement. 
He shows that the introduction of soap in the suspending fluid quickens and makes 
more persistent the movements of the suspended particles. Soap in the eyes of 
Professor Jevons acts conservatively by retaining or not conducting electricity. 
In my eyes it is a colloid, keeping up movements by revolutionary perturbations. \dots 
It is interesting to remember that, while soap is probably our best detergent, 
boiled oatmeal is one of its best substitutes. What this may be as a conductor of 
electricity I do not know, but it certainly is a colloid mixture or solution.''

Careful experiments and arguments supporting the kinetic theory were 
made by Gouy. From his work and the work of others emerged the following main 
points (cf. [6]):

\begin{enumerate}
\item The motion is very irregular, composed of translations and rotations, 
and the trajectory appears to have no tangent. 
\item Two particles appear to move independently, even when they approach one
another to within a distance less than their diameter. 
\item The motion is more active the smaller the particles.
\item The composition and density of the particles have no effect. 
\item The motion is more active the less viscous the fluid. 
\item The motion is more active the higher the temperature. 
\item The motion never ceases. 
\end{enumerate}

In discussing 1, Perrin mentions the mathematical existence of no\-where 
differentiable curves. Point 2 had been noticed by Brown, and it is a strong argument 
against gross mechanical explanations. Perrin points out that 6 (although true) had 
not really been established by observation, since for a given fluid the viscosity usually 
changes by a greater factor than the absolute temperature, so that the effect 5 
dominates 6. Point~7 was established by observing a sample over a period of twenty 
years, and by observations of liquid inclusions in quartz thousands of years old. This 
point rules out all attempts to explain Brownian motion as a non-equilibrium phenomenon.

By 1905, the kinetic theory, that Brownian motion of microscopic particles 
is caused by bombardment by the molecules of the fluid, seemed the most plausible. 
The seven points mentioned above did not seem to be in conflict with this theory. 
The kinetic theory appeared to be open to a simple test: the law of equipartition of 
energy in statistical mechanics implied that the kinetic energy of translation of a 
particle and of a molecule should be equal. The latter was roughly known (by a 
determination of Avogadro's number by other means), the mass of a particle could be 
determined, so all one had to measure was the velocity of a particle in Brownian motion. 
This was attempted by several experimenters, but the result failed to confirm the 
kinetic theory as the two values of kinetic energy differed by a factor of about 
$100{,}000$. The difficulty, of course, was point 1 above. What is meant by the 
velocity of a Brownian particle? This is a question that will recur throughout 
these lectures. The success of Einstein's theory of Brownian motion (1905) was 
largely due to his circumventing this question.

\begin{center}
\textrm{\bf References}
\end{center}

{\parskip=\smallskipamount

\noindent [6]. Jean Perrin, {\it Brownian movement and molecular reality}, translated from the 
Annales de Chimie et de Physique, $8^{\mathrm{me}}$ Series, 1909, by F. Soddy, Taylor and 
Francis, London, 1910.

\noindent [7]. William M. Ord, M.D., {\em On some Causes of Brownian Movements}, 
Journal of the Royal Microscopical Society, \textrm{2} (1879), 656--662. 
\medskip

The following also contain historical remarks (in addition to [6]). You 
are advised to consult at most one account, since they contradict each other not only 
in interpretation but in the spelling of the names of some of the people involved.

\noindent [8]. Jean Perrin, ``Atoms'', translated by D. A. Hammick, Van Nostrand, 1916. (Chapters III 
and IV deal with Brownian motion, and they are summarized in the author's article 
\emph{Brownian Movement} in the Encyclopaedia Britannica.)

\noindent [9]. E. F. Burton, {\em The Physical Properties of Colloidal Solutions}, Longmans, Green and Co., 
London, 1916. (Chapter IV is entitled \emph{The Brownian Movement}. Some of the physics in 
this chapter is questionable.)

\noindent [10]. Albert Einstein, {\em Investigations on the Theory of the Brownian Movement}, edited 
with notes by R. F\"{u}rth, translated by A. D. Cowper, Dover, 1956. (F\"{u}rth's first 
note, pp. 86--88, is historical.)

\noindent [11]. R. Bowling Barnes and S. Silverman, {\em Brownian Motion as a Natural Limit to all 
Measuring Processes}, Reviews of Modern Physics \textrm{6} (1934), 162--192.

}

\newpage

\fancyhead[CO]{ALBERT EINSTEIN}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Albert Einstein}

\hspace{6mm}It is sad to realize that despite all of the hard work that had gone 
into the study of Brownian motion, Einstein was unaware of the existence of the 
phenomenon. He predicted it on theoretical grounds and formulated a correct quantitative theory of it. (This was in 1905, the same year he discovered the special theory of 
relativity and invented the photon.) As he describes it [12, p.\ 47]:

``Not acquainted with the earlier investigations of Boltzmann and Gibbs, which had 
appeared earlier and actually exhausted the subject, I developed the statistical 
mechanics and the molecular-kinetic theory of thermodynamics which was based 
on the former. My major aim in this was to find facts which would guarantee as much 
as possible the existence of atoms of definite finite size. In the midst of this I 
discovered that, according to atomistic theory, there would have to be a movement 
of suspended microscopic particles open to observation, without knowing that 
observations concerning the Brownian motion were already long familiar.''

By the time his first paper on the subject was written, he had 
heard of Brownian motion [10, \S3, p.\ 1]:

``It is possible that the movements to be discussed here are identical with the 
so-called `Brownian molecular motion'; however, the information available to me 
regarding the latter is so lacking in precision, that I can form no judgment in the matter.''

There are two parts to Einstein's argument. The first is mathematical 
and will be discussed later (Chapter 5). The result is the following: Let $\rho = \rho (x,t)$ be 
the probability density that a Brownian particle is at $x$ at time $t$. Then, making 
certain probabilistic assumptions (some of them implicit), Einstein derived the 
diffusion equation
\begin{equation}
\frac{\partial \rho}{\partial t} = D \Delta \rho
\end{equation}
where $D$ is a positive constant, called the coefficient of diffusion. If the particle is at 0 
at time 0 $\big($so that $\rho (x,0) = \delta (x)$\big) then
\begin{equation}
\rho (x,t) = \frac{1}{(4 \pi D t)^{3/2}}\, e^{-\frac{|x|^2}{4Dt}} 
\end{equation}
(in three-dimensional space, where $|x|$ is the Euclidean distance of $x$ from the 
origin).

The second part of the argument, which relates $D$ to other physical 
quantities, is physical. In essence, it runs as follows. Imagine a suspension of 
many Brownian particles in a fluid, acted on by an external force $K$, and in 
equilibrium. (The force $K$ might be gravity, as in the figure, but the beauty of 
the argument is that $K$ is entirely virtual.)
$$\vbox{\pdfimage{1.png}}$$
$$\hbox{Figure 1}$$

In equilibrium, the force $K$ is balanced by the osmotic pressure forces 
of the suspension,
\begin{equation}
K = kT\,\frac{{\mathop{\rm grad} \nolimits} \,\nu}{\nu}. 
\end{equation}
Here $\nu$ is the number of particles per unit volume, $T$ is the absolute temperature, 
and $k$ is Boltzmann's constant. Boltzmann's constant has the dimensions of 
energy per degree, so that $kT$ has the dimensions of energy. A knowledge of $k$ is 
equivalent to a knowledge of Avogadro's number, and hence of molecular sizes. The right 
hand side of (4.3) is derived by applying to the Brownian particles the same considerations 
that are applied to gas molecules in the kinetic theory.

The Brownian particles moving in the fluid experience a resistance due to friction, and the force $K$ imparts to each particle a velocity of the form
\[
\frac{K}{m \beta},
\]
where $\beta$ is a constant with the dimensions of frequency (inverse time) and $m$ 
is the mass of the particle. Therefore
\[
\frac{\nu K}{m \beta}
\]
particles pass a unit area per unit of time due to the action of the force~$K$. On the 
other hand, if diffusion alone were acting, $\nu$ would satisfy the diffusion equation
\[
\frac{\partial \nu }{\partial t} = D \Delta \nu
\]
so that
\[
- D\, {\mathop{\rm grad} \nolimits} \,\nu
\]
particles pass a unit area per unit of time due to diffusion. In dynamical equilibrium, therefore,
\begin{equation}
\frac{\nu K }{m \beta} = D\, {\mathop{\rm grad} \nolimits} \,\nu.
\end{equation}
Now we can eliminate $K$ and $\nu$ between (4.3) and (4.4), giving Einstein's formula \\
\begin{equation}
D = \frac{k T}{m \beta}.
\end{equation}
This formula applies even when there is no force and when there is only one Brownian 
particle (so that $\nu$ is not defined).

Parenthetically, if we divide both sides of (4.3) by $m \beta$, and 
use (4.5), we obtain
\[
\frac{K}{m \beta} = D \frac{{\mathop{\rm grad} \nolimits} \,\nu}{\nu} .
\]
The probability density $\rho$ is just the number density $\nu$ divided by the 
total number of particles, so this can be rewritten as
\[
\frac{K}{m \beta} = D \frac{{\mathop{\rm grad} \nolimits} \,\rho}{\rho} .
\]
Since the left hand side is the velocity acquired by a particle due to the 
action of the force,
\begin{equation}
D \frac{{\mathop{\rm grad} \nolimits} \,\rho}{\rho} 
\end{equation}
is the velocity required of the particle to counteract osmotic effects.

If the Brownian particles are spheres of radius $a$, then Stokes' 
theory of friction gives $m \beta =6 \pi \eta a$, where $\eta$ is the coefficient of 
viscosity of the fluid, so that in this case
\begin{equation}
D = \frac{k T}{6 \pi \eta a} . 
\end{equation}
The temperature $T$ and the coefficient of viscosity $\eta$ can be measured, with 
great labor a colloidal suspension of spherical particles of fairly uniform radius $a$ 
can be prepared, and $D$ can be determined by statistical observations of Brownian 
motion using (4.2). In this way Boltzmann's constant $k$ (or, equivalently, Avogadro's 
number) can be determined. This was done in a series of difficult and laborious 
experiments by Perrin and Chaudesaigues [6, \S3]. Rather surprisingly, considering 
the number of assumptions that went into the argument, the result obtained for 
Avogadro's number agreed to within 19\% of the modern value obtained by other means. 
Notice how the points 3--6 of Chapter~3 are reflected in the formula (4.7).

Einstein's argument does not give a dynamical theory of Brownian 
motion; it only determines the nature of the motion and the value of the diffusion 
coefficient on the basis of some assumptions. Smoluchowski, independently of Einstein, 
attempted a dynamical theory, and arrived at~(4.5) with a factor of $32/27$ of the 
right hand side. Langevin gave another derivation of (4.5) which was the starting point 
for the work of Ornstein and Uhlenbeck, which we shall discuss later (Chapters 9--10). 
Langevin is the founder of the theory of stochastic differential equations (which is 
the subject matter of these lectures).

Einstein's work was of great importance in physics, for it showed in 
a visible and concrete way that atoms are real. Quoting from Einstein's 
Autobiographical Notes again [12, p.\ 49]:

``The agreement of these considerations with experience together with Planck's 
determination of the true molecular size from the law of radiation (for high 
temperatures) convinced the sceptics, who were quite numerous at that time 
(Ostwald, Mach) of the reality of atoms. The antipathy of these scholars towards 
atomic theory can indubitably be traced back to their positivistic philosophical attitude. 
This is an interesting example of the fact that even scholars of audacious spirit and fine 
instinct can be obstructed in the interpretation of facts by philosophical prejudices.'' 

Let us not be too hasty in adducing any other interesting example that may spring to mind. 

%~ \\[0.5cm]
\begin{center}
\textrm{\bf Reference}
\end{center}
[12]. Paul Arthur Schilpp, editor, ``Albert Einstein: Philosopher-Scien\-tist'', The Library 
of Living Philosophers, Vol.\ VII, The Library of Living Philosophers, Inc., Evanston, 
Illinois, 1949.

\newpage
\thispagestyle{empty}

\fancyhead[CO]{DERIVATION OF THE WIENER PROCESS}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}


\chapter{Derivation of the Wiener process}

\hspace{6mm}Einstein's basic assumption is that the following is possible [10, \S3, p.~13]: 
``We will introduce a time-interval $\tau$ in our discussion, 
which is to be very small compared with the observed interval of time 
[i.e., the interval of time between observations], but, nevertheless,
of such a magnitude that the movements executed by a particle 
in two consecutive intervals of time $\tau$ are to be considered 
as mutually independent phenomena.'' 
He then implicitly considers the limiting case $\tau \to 0$. 
This assumption has been criticized by many people, including Einstein himself, 
and later on (Chapter 9--10) we shall discuss a theory in which this assumption is modified. 
Einstein's derivation of the transition probabilities proceeds by formal manipulations of power series. His neglect of higher order terms is tantamount to the assumption (5.2) below. 
In the theorem below, $p^t$ may be thought of as the probability distribution 
at time $t$ of the $x$-coordinate of a Brownian particle starting at $x=0$ at $t=0$. 
The proof is taken from a paper of Hunt [13], 
who showed that Fourier analysis is not the natural tool for problems of this type.
\bigskip

%\begin{thm}
T{\ninerm HEOREM} 5.1 {\it
Let $p^t$, $0\leq t<\infty $, be a family of probability measures on the real line $\Bbb{R}$ such that 
\begin{equation}
p^t *p^s =p^{t+s}; \qquad \qquad 0\leq t,s<\infty,
\end{equation}
where $*$ denotes convolution; for each $\varepsilon >0$,
\begin{equation}
p^t (\{ x:|x| \geq \varepsilon \}) = {\rm o} (t), \qquad \qquad t \to 0;
\end{equation}
and for each $t>0$, $p^t$ is invariant under the transformation $x \mapsto -x$. 
Then either $p^t =\delta$ for all $t \geq 0$ or there is a $D>0$ such that, 
for all $t>0$, $p^t$ has the density 
\[
p(t,x) = \frac{1}{\sqrt{4 \pi D t}} e^{-\frac{x^2}{4Dt}},
\]
so that $p$ satisfies the diffusion equation
\[
\frac{\partial p}{\partial t} = D \frac{\partial ^2 p}{\partial x^2}, \qquad \qquad t>0.
\]
}
%\end{thm}

First we need a lemma:
\\

%\begin{thm}
T{\ninerm HEOREM} 5.2 {\it
Let $\mathscr{X}$ be a real Banach space, $f\in \mathscr{X}$, 
$ \mathscr{D}$ a dense linear subspace of $\mathscr{X}$, $u_1, \ldots, u_n$ 
continuous linear functionals on $\mathscr{X}$, $ \delta >0$. 
Then there exists a $g \in \mathscr{D}$ with
\[
\| f-g \| \leq \delta
\]
\vskip-30pt
\[
(u_1,f)=(u_1,g), \ldots, (u_n,f)=(u_n,g).
\]
}
%\end{thm}

{\it Proof.} Let us instead prove that if $ \mathscr{X}$ is a real Banach space, 
$\mathscr{D}$ a dense convex subset, $\mathscr{M}\,$a closed affine hyperplane, 
then $ \mathscr{D} \cap \mathscr{M}$ is dense in $ \mathscr{M}$. 
Then the general case of finite co-dimension follows by induction.

Without loss of generality, we can assume that $ \mathscr{M}$ is linear 
$(0 \in \mathscr{M})$, so that, if we let $e$ be an element of $ \mathscr{X}$ not in $ \mathscr{M}$,
\[
\mathscr{X} = \mathscr{M} \oplus \Bbb{R} e.
\]

Let $f \in \mathscr{M}$, $\varepsilon > 0$. Choose $g_+$ in 
$ \mathscr{D}$ so that
\[
\| (f + e)-g_+ \| \leq \varepsilon
\]
and choose $g_-$ in $ \mathscr{D}$ so that
\[
\| (f-e)-g_- \| \leq \varepsilon .
\]
Set
\begin{align*}
g_+ &= m_+ + r_+ e,\quad m_+ \in \mathscr{M} \\
g_- &= m_- + r_- e,\quad m_- \in \mathscr{M}.
\end{align*}
Since $ \mathscr{M}$ is closed, the linear functional that assigns to each element of~$ \mathscr{X}$ 
the corresponding coefficient of $e$ is continuous. 
Therefore $r_+ $ and $r_-$ tend to $1$ as $\varepsilon \to 0$ and so are strictly positive 
for $\varepsilon$ sufficiently small. By the convexity of $ \mathscr{D}$,
\[
g =\frac{r_- g_+ + r_+ g_-}{r_- + r_+}
\]
is then in $ \mathscr{D}$. But
\[
g =\frac{r_- m_+ + r_+ m_-}{r_- + r_+}
\]
is also in $ \mathscr{M}$, and it converges to $f$ as $ \varepsilon \to 0$. ~~\textrm{QED}.
\\

We recall that if $ \mathscr{X}$ is a Banach space, then a 
\emph{contraction semigroup} on~$ \mathscr{X}$ (in our terminology) is a family of bounded linear
transformations~$P^t$ of $ \mathscr{X}$ into itself, defined for $0 \le t < \infty$, such 
that $P^0 = 1$, $P^t P^s = P^{t+s}$, $\| P^t f-f \| \to 0$, and 
$\| P^t \| \leq 1$, for all $0 \leq t$, $s< \infty$ and all~$f$ in $ \mathscr{X}$. 
The \emph{infinitesimal generator} $A$ is defined by
\[
A f = \lim_{t \to 0^+} \frac{P^t f -f}{t}
\]
on the domain $ \mathscr{D} (A)$ of all $f$ for which the limit exists.

If $X$ is a locally compact Hausdorff space, $C(X)$ denotes the 
Banach space of all continuous functions vanishing at infinity in the norm
\[
\| f \| = \sup_{x \in X} |f(x)|,
\]
and $ \dot{X}$ denotes the one-point compactification of $X$. We denote by 
$C_{{\rm com}}^{2} (\Bbb{R}^{\ell})$ the set of all functions of class $C^2$ with compact support 
on~$\Bbb{R}^{\ell}$, by $C^2(\Bbb{R}^{\ell})$ its completion in the norm
\[
\| f \|^{\dagger} = \| f \| + \sum_{i=1}^{\ell} \| \frac{\partial f}{\partial x^i} \| + 
\sum_{i,j=1}^{\ell} \| \frac{\partial^2 f}{\partial x^i \partial x^j} \|,
\]
and by $C^2 (\dot{\Bbb{R} }^{\ell})$ the completion of $C_{{\rm com}}^{2} (\dot{\Bbb{R} }^{\ell})$ 
together with the constants, in the same norm.

A \emph{Markovian semigroup} on $C(X)$ is a contraction semigroup 
on $C(X)$ such that $f \ge 0$ implies $P^t f \ge 0 $ for $0 \le t < \infty$, and such 
that for all $x$ in $X$ and $0 \le t<\infty$,
\[
\sup_{\substack{0 \leq f \leq 1 \\ f \in C(X) }} P^t f(x) = 1.
\]
If $X$ is compact, the last condition is equivalent to $P^t 1 = 1$, $0 \leq t <\infty$.
By the Riesz theorem, there is a unique regular Borel probability measure $p^t (x,\cdot)$ 
such that
\[
P^t f(x) = \int f(y)\,p^t(x,dy) ,
\]
and $p^t$ is called the \emph{kernel} of $P^t$.
\\

%\begin{thm}
T{\ninerm HEOREM} 5.3 {\it
Let $P^t$ be a Markovian semigroup on $C (\dot{\Bbb{R} }^{\ell})$ commuting with 
translations, and let $A$ be the infinitesimal generator of $P^t$. Then
\[
C^2 (\dot{\Bbb{R} }^{\ell}) \subseteq \mathscr{D}(A).
\]
}
%\end{thm}

{\it Proof.} Since $P^t$ commutes with translations, $P^t$ leaves 
$C^2 (\dot{\Bbb{R} }^{\ell})$ invariant and is a contraction semigroup on it. Let $A^{\dag}$
be the infinitesimal generator of $P^t$ on $C^2 (\dot{\Bbb{R} }^{\ell})$. Clearly 
$ \mathscr{D} (A^{\dag}) \subseteq \mathscr{D} (A)$, and since the domain of the
infinitesimal generator is always dense, $ \mathscr{D}(A) \cap C^2 (\dot{\Bbb{R} }^{\ell})$
is dense in $C^2 (\dot{\Bbb{R} }^{\ell})$.

Let $\psi$ be in $C^2 (\dot{\Bbb{R} }^{\ell})$ and such that $\psi (x)=|x|^2$
in a neighborhood of~0, $\psi (x) =1$ in a neighborhood of infinity, and $\psi$ is strictly positive 
on $\dot{\Bbb{R} }^{\ell} -\{0\}$. Apply Theorem 5.2 to $ \mathscr{X} =
C^2 (\dot{\Bbb{R} }^{\ell})$, $ \mathscr{D} = \mathscr{D} (A) \cap C^2 (\dot{\Bbb{R} }^{\ell})$,
$f=\psi$, and to the continuous linear functionals mapping $\varphi$ in 
$ \mathscr{X}$ to
\[
\varphi (0), \; \frac{\partial \varphi}{\partial x^i} (0), \; \frac{\partial ^2 \varphi}
{\partial x^i \partial x^j} (0).
\]
Then, for all $\varepsilon > 0$, there is a $\varphi$ in $ \mathscr{D} (A)\cap 
C^2 (\dot{\Bbb{R} }^{\ell})$ with
\[
\varphi (0)= \frac{\partial \varphi}{\partial x^i} (0)=0, \; \frac{\partial ^2 \varphi}
{\partial x^i \partial x^j} (0)=2 \delta _{ij},
\]
and $\| \varphi-\psi \| \leq \varepsilon $. If $\varepsilon$ is small enough, $\varphi$ must
be strictly positive on $\dot{\Bbb{R} }^{\ell} -\{0\}$. Fix such a $\varphi$, and let $\delta >0$, 
$f \in C^2 (\dot{\Bbb{R} }^{\ell})$. By Theorem 5.2 again there is a $g$ in 
$\mathscr{D} (A) \cap C^2 (\dot{\Bbb{R} }^{\ell})$ with
\[
| f(y)-g(y) | \leq \delta \varphi (y) 
\]
for all $y$ in $\dot{\Bbb{R} }^{\ell}$. Now
\[
\frac{1}{t} \int | f(y)-g(y) |\, p^t (0,dy) \leq \frac{\delta}{t} \int \varphi (y) \,p^t (0,dy)
\]
and since $\varphi \in \mathscr{D} (A)$ with $\varphi (0)=0$, the right hand side is 
${\rm O} (\delta)$. Therefore
\begin{equation}
\frac{1}{t} \int [ f(y)-f(0) ] \,p^t (0,dy)
\end{equation}
and
\begin{equation}
\frac{1}{t} \int [ g(y)-g(0) ] \,p^t (0,dy)
\end{equation}
differ by ${\rm O} (\delta)$. Since $g \in \mathscr{D} (A)$, (5.4) has a limit as
$t \to 0$. Since $\delta $~is arbitrary, (5.3) has a limit as $t \to 0$. Therefore (5.3) 
is bounded as $t \to 0$. Since this is true for each $f$ in the Banach space 
$C^2 (\dot{\Bbb{R} }^{\ell})$, by the principle of uniform boundedness there is a constant
$K$ such that for all $f$ in $C^2 (\dot{\Bbb{R} }^{\ell})$ and $t > 0$,
\[
\left| \frac{1}{t} (P^t f-f)(0) \right| \leq K \|f\| ^{\dag}.
\]
By translation invariance,
\[
\left| \frac{1}{t} (P^t f-f)\ \right| \leq K \|f\| ^{\dag}.
\]
Now $\frac{1}{t} (P^t g-g) \to A g$ for all $g$ in the dense set $ \mathscr{D} (A^{\dag})
\cap C^2 (\dot{\Bbb{R} }^{\ell})$, so by the Banach-Steinhaus theorem, $\frac{1}{t} (P^t f-f)$
converges in $C (\dot{\Bbb{R} }^{\ell})$ for all $f$ in $C^2 (\dot{\Bbb{R} }^{\ell})$. ~~\textrm{QED}.
\\

%\begin{thm}
T{\ninerm HEOREM} 5.4 {\it
Let $P^t$ be a Markovian semigroup on $C (\dot{\Bbb{R} }^{\ell})$, not necessarily commuting 
with translations, such that $C^2_{{\rm com}} ({\Bbb{R} }^{\ell}) \subseteq \mathscr{D} (A)$, 
where $A$ is the infinitesimal generator of $P^t$. If for all $x$ in $\Bbb{R} ^{\ell}$ and 
all $\varepsilon > 0$
\begin{equation}
p^t (x,\{y: | y-x| \geq \varepsilon \}) = {\rm o}(t),
\end{equation}
then
\begin{equation}
Af(x) = \sum ^{\ell}_{i=1} b^i (x) \frac{\partial}{\partial x^i} f(x) + 
\sum ^{\ell} _{i,j=1} a^{ij} (x) \frac{\partial ^2}{\partial x^i \partial x^j} f(x)
\end{equation}
for all $f$ in $C^2 _{{\rm com}} ({\Bbb{R} }^{\ell})$, where the $a^{ij}$ and $b^i$ are real and 
continuous, and for each $x$ the matrix $a^{ij} (x) $ is of positive type.
}
%\end{thm}
\\

A matrix $a^{ij}$ is of {\it positive type} (positive definite, positive 
semi-definite, non-negative definite, etc.) in case for all complex $\zeta _i$,
\[
\sum ^{\ell} _{i,j=1} \bar{\zeta _i} a^{ij} \zeta_j \geq 0.
\]
The operator $A$ is not necessarily elliptic since the matrix $a^{ij} (x)$ may be singular. 
If $P^t$ commutes with translations then $a^{ij}$ and $b^i$ are constants, of course.\\

{\it Proof.} Let $f \in C^2_{{\rm com}} (\Bbb{R} ^{\ell})$ and suppose that 
$f$ together with its first and second order partial derivatives vanishes at $x$. 
Let $g \in C^2_{{\rm com}} (\Bbb{R} ^{\ell})$ be such that $g(y) =|y-x| ^2$ in a neighborhood of 
$x$ and $g \geq 0$. Let $\varepsilon >0$ and let $U=\{y:|f(y)|\leq \varepsilon g(y)\} $,
so that $U$ is a neighborhood of $x$. By~(5.5), $p^t (x,\Bbb{R}^{\ell}\setminus U) = {\rm o}(t)$ and so
\begin{eqnarray*}
A f(x) &=& \lim_{t \to 0} \frac{1}{t} \int f(y) p^t (x,dy) \\
&=& \lim_{t \to 0} \frac{1}{t} \int _{U} f(y) p^t (x,dy) \leq \varepsilon \lim _{t \to 0} 
\frac{1}{t} \int _{U} g(y) p^t (x,dy) = \varepsilon A g(x).
\end{eqnarray*}
Since $\varepsilon$ is arbitrary, $Af(x)=0$. This implies that $Af(x)$ is of the form~(5.6)
for certain real numbers $a^{ij} (x)$, $b^i (x)$, and we can assume that the $a^{ij} (x)$ 
are symmetric. (There is no zero-order term since $P^t$ is Markovian.) If we apply $A$ 
to functions in $C^2_{{\rm com}} (\Bbb{R} ^{\ell})$ that in a neighborhood of $x$ agree with 
$y^i-x^i $ and $(y^i-x^i)(y^j-x^j)$, we see that $b^i$ and $a^{ij}$ are continuous. 
If $f$ is in $C^2_{{\rm com}} (\Bbb{R} ^{\ell})$ and $f(x)=0$ then
\[
A f^2 (x) = \lim _{t \to 0} \frac{1}{t} \int f^2 (y) p^t (x,dy) \geq 0.
\]
Therefore
\begin{eqnarray*}
A f^2 (x) & =& \sum ^{\ell} _{i,j=1} a^{ij} (x) \frac{\partial ^2 f^2}{\partial x^i \partial x^j} (x)\\
&=& 2 \sum ^{\ell} _{i,j=1} a^{ij} (x) \frac{\partial f}{\partial x^i} (x) 
\frac{\partial f}{\partial x^j} (x) \geq 0.
\end{eqnarray*}
We can choose
\[
\frac{\partial f}{\partial x^i } (x) = \xi ^i
\]
to be arbitrary real numbers, and since $a^{ij} (x)$ is real and symmetric, $a^{ij} (x)$ 
is of positive type. ~~\textrm{QED}.
\\

%\begin{thm}
T{\ninerm HEOREM} 5.5 {\it
Let $P^t$ be a Markovian semigroup on $C (\Bbb{R} ^{\ell})$ commuting with translations, and 
let $A$ be its infinitesimal generator. Then
\begin{equation}
C^2 (\Bbb{R} ^{\ell}) \subseteq \mathscr{D} (A)
\end{equation}
and $P^t$ is determined by $A$ on $C^2_{{\rm com}} (\Bbb{R} ^{\ell})$.
}
%\end{thm}
\\

{\it Proof.} The inclusion (5.7) follows from Theorem 5.3. The proof 
of that theorem shows that $A$ is continuous from $C^2_{{\rm com}} (\Bbb{R} ^{\ell})$ into 
$C (\Bbb{R} ^{\ell})$, so that $A$ on $C^2_{{\rm com}} (\Bbb{R} ^{\ell})$ determines $A$ on
$C^2 (\Bbb{R} ^{\ell})$ by continuity. Since $P^t$ commutes with translations, $P^t$ 
leaves $C^2 (\Bbb{R} ^{\ell})$ invariant.

Let $\lambda >0$. We shall show that $(\lambda -A)C^2 (\Bbb{R} ^{\ell})$
is dense in $C (\Bbb{R} ^{\ell})$. Suppose not. Then there is a non-zero continuous 
linear functional $z$ on~$C (\Bbb{R} ^{\ell})$ such that $\big(z,(\lambda -A)f\big)=0$ for all 
$f$ in $C^2 (\Bbb{R} ^{\ell})$. Since $C^2 (\Bbb{R} ^{\ell})$ is dense in $C (\Bbb{R} ^{\ell})$, 
there is a $g$ in $C^2 (\Bbb{R} ^{\ell})$ with $(z,g) \ne 0$. Then
\[
\frac{d}{d t} (z, P^t g) = (z,A P^t g) = (z, \lambda P^t g) = \lambda (z,P^t g)
\]
since $P^t g$ is again in $C^2 (\Bbb{R} ^{\ell})$. Therefore
\[
(z,P^t g) = e^{\lambda t} (z,g)
\]
is unbounded, which is a contradiction. It follows that if $Q^t$ is another such 
semigroup with infinitesimal generator $B$, and $B=A$ on $C^2_{{\rm com}} (\Bbb{R} ^{\ell})$, 
then $(\lambda -B)^{-1} = (\lambda -A)^{-1}$ for $\lambda >0$. But these are the 
Laplace transforms of the semigroups $Q^t$ and $P^t$, 
and by the uniqueness theorem for Laplace transforms, 
$Q^t=P^t$. ~~\textrm{QED}. \\

Theorem 5.1 follows from theorems 5.3, 5.4, 5.5 and the well-known formula 
for the fundamental solution of the diffusion equation.

%~ \\[1.5cm]
\begin{center}
\textrm{\bf References}
\end{center}
[13]. G. A. Hunt, {\em Semi-group of measures on Lie groups}, Transactions of the 
American Mathematical Society \textrm{81} (1956), 264--293.

\noindent (Hunt treats non-local processes as well, on arbitrary Lie groups.)
\smallskip

Banach spaces, the principle of uniform boundedness, the 
Banach-Steinhaus theorem, semigroups and infinitesimal generators are all 
discussed in detail in:
\smallskip

\noindent [14]. Einar Hille and Ralph S. Phillips, ``Functional Analysis and Semi-Groups'', 
revised edition, American Math.\ Soc.\ Colloquium Publications, vol.\ XXXI, 1957.

%\newpage
%\thispagestyle{empty}
~

\newpage

\fancyhead[CO]{GAUSSIAN PROCESSES}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Gaussian processes}

\hspace{6mm}Gaussian random variables were discussed by Gauss in 1809 and 
the central limit theorem was stated by Laplace in 1812. Laplace had already 
considered Gaussian random variables around 1780, and for this reason Frenchmen 
call Gaussian random variables ``Laplacian''. However, the Gaussian measure 
and an important special case of the central limit theorem were discovered by 
de Moivre in 1733. The main tool in de Moivre's work was Stirling's formula, 
which, except for the fact that the constant occurring in it is $\sqrt{2 \pi}$, 
was discovered by de Moivre. In statistical mechanics the Gaussian distribution 
is called ``Maxwellian''. Another name for it is ``normal''.

A \emph{Gaussian measure} on $\Bbb{R}^{\ell}$ is a measure that 
is the transform of the measure with density
\[
\frac{1}{(2 \pi)^{\ell /2}} e^{-\frac{1}{2} |x|^2}
\]
under an affine transformation. It is called \emph{singular} in case the affine 
transformation is singular, which is the case if and only if it is singular with respect 
to Lebesgue measure.

A set of random variables is called Gaussian in case the distribution of 
each finite subset is Gaussian. A set of linear combinations, or limits in measure of 
linear combinations, of Gaussian random variables is Gaussian. Two (jointly) Gaussian 
random variables are independent if and only if they are uncorrelated; i.e., 
their covariance
\[
r(x,y) = \mathrm{E} (x- \mathrm{E}x) (y-\mathrm{E}y) 
\]
is zero (where $\mathrm{E}$ denotes the expectation).

We define the \emph{mean} $m$ and \emph{covariance} $r$ of 
a probability measure $\mu$ on $\Bbb{R}^{\ell}$ as follows, provided the integrals exist:
\[
m_i = \int x_i \mu (dx)
\]
\[
r_{ij} = \int x_i x_j \mu (dx)-m_i m_j = \int (x_i-m_i) (x_j-m_j) \mu (dx)
\]
where $x$ has the components $x_i$. The covariance matrix $r$ is of positive type. 
Let $\mu$ be a probability measure on $\Bbb{R}^{\ell}$, $\hat{\mu}$ its inverse 
Fourier transform
\[
\hat{\mu} (\xi) = \int e^{i \xi \cdot x} \mu (dx).
\]
Then $\mu$ is Gaussian if and only if
\[
\hat{\mu} (\xi) = e^{-\frac{1}{2} \sum r_{ij} \xi_i \xi_j + i \sum m_i \xi_i}
\]
in which case $r$ is the covariance and $m$ the mean. If $r$ is nonsingular and 
$r^{-1}$ denotes the inverse matrix, then the Gaussian measure with mean $m$ and 
covariance $r$ has the density
\[
\frac{1}{(2 \pi)^{\ell /2} (\det r)^{\frac{1}{2}}} e^{-\frac{1}{2} 
\sum (r^{-1}) _{ij} (x_i -m_i)(x_j-m_j)}.
\]
If $r$ is of positive type there is a unique Gaussian measure with covariance~$r$ 
and mean $m$.

A set of complex random variables is called Gaussian if and only if 
the real and imaginary parts are (jointly) Gaussian. We define the covariance of
complex random variables by
\[
r(x,y) = \mathrm{E} (\bar{x} -\mathrm{E} \bar{x}) (y -\mathrm{E} y).
\]

Let $T$ be a set. A complex function $r$ on $ T \times T$ is called of 
{\it positive type}
in case for all $ t_1, \ldots, t_{\ell}$ in $T$ the matrix $r(t_i,t_j)$ is of 
positive type. Let $x$ be a stochastic process indexed by $T$. We call 
$r(t,s) =r\big(x(t),x(s)\big)$ the \emph{covariance} of the process, $m(t) =\mathrm{E} x(t)$ 
the \emph{mean} of the process (provided the integrals exist). The covariance is 
of positive type.

The following theorem is immediate (given the basic existence 
theorem for stochastic processes with prescribed finite joint distributions).
\\

%\begin{thm}
T{\ninerm HEOREM} 6.1 {\it
Let $T$ be a set, $m$ a function on $T$, $r$ a function of positive type on $T \times T$. 
Then there is a Gaussian stochastic process indexed by $T$ with mean $m$ and 
covariance $r$. Any two such are equivalent. 
}
%\end{thm}

%~ \\[0.5cm]
\begin{center}
\textrm{\bf Reference}
\end{center}
[15]. J. L. Doob, ``Stochastic Processes'', John Wiley \& Sons, Inc., New York, 1953. 
(Gaussian processes are discussed on pp.\ 71--78.)

\newpage
\thispagestyle{empty}
~

%\newpage
%\thispagestyle{empty}
%~


\newpage

\fancyhead[CO]{THE WIENER INTEGRAL}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{The Wiener integral}

\hspace{6mm}The differences of the Wiener process
\[
w(t) -w(s), \quad \quad 0 \leq s \leq t < \infty
\]
form a Gaussian stochastic process, indexed by pairs of positive numbers $s$ and $t$ with $s\le t$.
This difference process has mean $0$ and covariance
\[
\mathrm{E} \big(w(t) -w(s)\big) \big(w(t ')-w(s ')\big) = \sigma ^2 \,| [ s,t] \cap [s ',t '] |
\]
where $|~|$ denotes Lebesgue measure, and $\sigma ^2$ is the variance 
parameter of the Wiener process.

We can extend the difference process to all pairs of real numbers 
$s$ and $t$. We can arbitrarily assign a distribution to $w(0)$. The resulting 
stochastic process $w(t)$, $- \infty < t < \infty$, is called the two-sided Wiener process. 
It is Gaussian if and only if $w(0)$ is Gaussian (e.g., $w(0)=x_0$ where $x_0$ is a fixed 
point), but in any case the differences are Gaussian. If we know that a Brownian 
particle is at $x_0$ at the present moment, $w(0)=x_0$, then $w(t)$ for $t >0$ is the 
position of the particle at time $t$ in the future and $w(t)$ for $t <0$ is the position of the 
particle at time $t$ in the past. A movie of Brownian motion looks, statistically, the same 
if it is run backwards.

We recall that, with probability one, the sample paths of the Wiener 
process are continuous but not differentiable. Nevertheless, integrals of the form
\[
\int ^{\infty} _{- \infty} f(t)\, d w(t) 
\]
can be defined, for any square-integrable $f$.
\\

%\begin{thm}
T{\ninerm HEOREM} 7.1 {\it
Let $\Omega $ be the probability space of the differences of the two-sided Wiener 
process. There is a unique isometric operator from $L^2( \Bbb{R}, \sigma ^2 dt) $ 
into $L^2 (\Omega)$, denoted by
\[
f \mapsto \int ^{\infty}_{- \infty} f(t)\, d w(t), 
\]
such that for all $- \infty < a \leq b < \infty$,
\[
\int ^{\infty} _{- \infty} \chi_{[a,b]} (t)\, d w(t) = w(b) -w(a).
\]
The set of $\int ^{\infty} _{- \infty} f(t)\, d w(t) $ is Gaussian.
}
\\
%\end{thm}

If $E$ is any set, $\chi _E$ is its characteristic function,
\[
\chi_E (t) =
\begin{cases}
1, & t \in E \\
0, & t \not\in E. 
\end{cases}
\]
We shall write, in the future, $\int ^b _a f(t) \,dw(t) $ for $\int ^{\infty} _{- \infty} 
\chi _{[a,b]} (t)f(t)\, d w(t)$. \\

{\it Proof.} Let $f$ be a step function
\[
f = \sum ^n_{i=1} c_i \chi _{[a_i,b_i]} .
\]
Then we define
\begin{equation}
\int ^{\infty} _{- \infty} f(t)\, d w(t) = \sum ^n _{i =1} c_i [w(b_i) -w(a_i) ].
\end{equation}
If $g$ also is a step function,
\[
g = \sum ^m _{j=1} d_j \chi_{[ e_j, f_j]}, 
\]
then
\begin{eqnarray*}
\lefteqn{ \mathrm{E} \int ^{\infty} _{- \infty} f(t)\, d w(t) 
\int ^{\infty} _{- \infty} g(s)\, d w(s) } \\
&=& \mathrm{E} \sum ^n _{i =1} c_i [w(b_i) -w(a_i) ] 
\sum ^m _{j=1} d_j [w(f_j) -w(e_j)] \\
&=& \sum ^n _{i =1} \sum ^m _{j=1} c_i d_j \sigma ^2 \,| [w(b_i) -w(a_i) ] 
\cap [w(f_j) -w(e_j)] | \\
&=& \sigma ^2 \int ^{\infty} _{- \infty} f(t) g(t)\, dt .
\end{eqnarray*}
Since the step functions are dense in $L^2 ( \Bbb{R}, \sigma ^2 dt)$, the mapping 
extends by continuity to an isometry. Uniqueness is clear, and so is the fact that 
the random variables are Gaussian. ~~\textrm{QED}. \\

The Wiener integral can be generalized. Let $T$, $\mu$ be an arbitrary 
measure space, and let $\mathscr{S}_0$ denote the family of measurable sets of 
finite measure. Let $w$ be the Gaussian stochastic process indexed by $\mathscr{S}_0$ 
with mean $0$ and covariance $r(E,F) = \mu (E \cap F)$. This is easily seen to be of 
positive type (see below). Let $\Omega$ be the probability space of the $w$-process.
\\

%\begin{thm}
T{\ninerm HEOREM} 7.2 {\it
There is a unique isometric mapping
\[
f \mapsto \int f(t) \,dw(t)
\]
from $L^2 (T,\mu) $ into $L^2 (\Omega)$ such that, for $E \in \mathscr{S}_0$,
\[
\int \chi _E (t) \,dw(t) = w(E).
\]
The $\int f(t) \,dw(t) $ are Gaussian.
}
\\
%\end{thm}

The proof is as before.

If $\mathscr{H}$ is a Hilbert space, the function $r$ on 
$ \mathscr{H} \times \mathscr{H}$ that is the inner product, $r(f,g) = (f,g)$, is of 
positive type, since
\[
\sum \bar{\zeta _i} (f_i, f_j) \zeta _j = \| \sum _j \zeta _j f_j \|^2 \geq 0.
\]
Consequently, the Wiener integral can be generalized further, as a purely Hilbert 
space theoretic construct.
\\

%\begin{thm}
T{\ninerm HEOREM} 7.3 {\it
Let $\mathscr{H}$ be a Hilbert space. Then there is a Gaussian stochastic process, 
unique up to equivalence, with mean $0$ and covariance given by the inner product.
}
\\
%\end{thm}

{\it Proof.} This follows from Theorem 6.1. ~~\textrm{QED}. \\

The special feature of the Wiener integral on the real line that 
makes it useful is its relation to differentiation.
\\

%\begin{thm}
T{\ninerm HEOREM} 7.4 {\it
Let $f$ be of bounded variation on the real line with compact support, and let 
$w$ be a Wiener process. Then
\begin{equation}
\int ^{\infty} _{- \infty} f(t) \,dw(t) =-\int ^{\infty} _{- \infty} df(t)\,w(t).
\end{equation}
In particular, if $f$ is absolutely continuous on $[a,b]$, then
\[
\int ^b _a f(t)\,dw(t) =-\int ^b_a f' (t) w(t)\,dt +f(b)w(b)-f(a)w(a).
\]
}
\\
%\end{thm}

The left hand side of (7.2) is defined since $f$ must be in $L^2$. 
The right hand side is defined a.e. (with probability one, that is) since almost every 
sample function of the Wiener process is continuous. The equality in (7.2) means 
equality a.e., of course. \\

{\it Proof.} If $f$ is a step function, (7.2) is the definition (7.1) 
of the Wiener integral. In the general case we can let $f_n$ be a sequence of step functions
such that $f_n \rightarrow f$ in $L^2$ and $df_n \rightarrow df$ in the weak-$*$ topology of measures,
so that we have convergence to the two sides of (7.2). ~~\textrm{QED}.

%~ \\[1cm]
\begin{center}
\textrm{\bf References}
\end{center}

See Doob's book [15, \S6, p. 426] for a discussion of the Wiener integral. 
The purely Hilbert space approach to the Wiener integral, together with applications, 
has been developed by Irving Segal and others. See the following and its bibliography:
\smallskip

\noindent [16]. Irving Segal, \emph{Algebraic integration theory}, Bulletin American 
Math.\ Soc.\ \textrm{71} (1965), 419--489.
\smallskip

For discussions of Wiener's work see the special commemorative issue:

\smallskip

\noindent [17]. Bulletin American Math.\ Soc.\ \textrm{72} (1966), No. 1 Part 2.
\smallskip

We are assuming a knowledge of the Wiener process. For an exposition 
of the simplest facts, see Appendix A of:
\smallskip

\noindent[18]. Edward Nelson, \emph{Feynman integrals and the Schr\"{o}dinger equation}, Journal 
of Mathematical Physics \textrm{5} (1964), 332--343.
\smallskip

For an account of deeper facts, see:
\smallskip

\noindent [19]. Kiyosi It\^{o} and Henry P. McKean, Jr., ``Diffusion Processes and their 
Sample Paths'', Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen 
vol.\ 125, Academic Press, Inc., New York, 1965.

\newpage
\thispagestyle{empty}
~

\newpage

\fancyhead[CO]{A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{A class of stochastic differential equations}

\hspace{6mm}By a Wiener process on $\Bbb{R} ^{\ell}$ we mean a Markov process 
$w$ whose infinitesimal generator $C$ is of the form
\begin{equation}
C = \sum ^{\ell} _{i,j=1} c^{ij} \frac{\partial ^2}{\partial x^i \partial x^j},
\end{equation}
where $c^{ij}$ is a constant real matrix of positive type. Thus the $w(t)-w(s)$ are 
Gaussian, and independent for disjoint intervals, with mean 0 and covariance matrix 
$2 c^{ij} |t-s|$.
\\

%\begin{thm}
T{\ninerm HEOREM} 8.1 {\it
Let $b:\Bbb{R} ^{\ell} \to \Bbb{R} ^{\ell}$ satisfy a global Lipschitz condition; 
that is, for some constant $\kappa$,
\[
| b(x_0) -b(x_1) | \leq \kappa |x_0-x_1| 
\]
for all $x_0$ and $x_1$ in $\Bbb{R} ^{\ell}$. Let $w$ be a Wiener process on 
$\Bbb{R} ^{\ell}$ with infinitesimal generator $C$ given by {\rm (8.1)}. For each $x_0$ 
in $\Bbb{R} ^{\ell}$ there is a unique stochastic process $x(t)$, $0 \leq t < \infty$, 
such that for all $t$
\begin{equation}
x(t) = x_0 + \int ^t _0 b\big(x(s)\big)\,ds + w(t)-w(0) .
\end{equation}
The $x$ process has continuous sample paths with probability one.

If we define $P^t f(x_0) $ for $0 \leq t < \infty$, $x_0 \in 
\Bbb{R} ^{\ell}$, $f \in C(\Bbb{R} ^{\ell})$ by
\begin{equation}
P^t f(x_0) = \mathrm{E} f\big(x(t)\big),
\end{equation}
where $\mathrm{E}$ denotes the expectation on the probability space of the $w$ process, 
then $P^t$ is a Markovian semigroup on $C(\Bbb{R} ^{\ell})$. Let $A$ be the 
infinitesimal generator of $P^t$. Then $C^2 _{{\rm com}} (\Bbb{R} ^{\ell}) \subseteq 
\mathscr{D} (A)$ and
\begin{equation}
Af = b \cdot \nabla f + Cf
\end{equation}
for all $f$ in $C^2 _{{\rm com}} (\Bbb{R} ^{\ell})$.
}
%\end{thm}
\\

{\it Proof.} With probability one, the sample paths of the $w$ 
process are continuous, so we need only prove existence and uniqueness for (8.2) with 
$w$ a fixed continuous function of $t$. This is a classical result, even when $w$~is not 
differentiable, and can be proved by the Picard method, as follows.

Let $\lambda >\kappa$, $t \geq 0$, and let $\mathscr{X}$ be the Banach 
space of all continuous functions $\xi$ from $[0,t]$ to $\Bbb{R} ^{\ell}$ with the norm 
\[
\| \xi \| = \sup _{0 \leq s \leq t} e^{- \lambda s} |\xi (s)| .
\]
Define the non-linear mapping $T : \mathscr{X} \to \mathscr{X}$ by
\[
T \xi (s) = \xi (0) + \int ^s _0 b\big(\xi (r)\big)\, dr + w(s) -w(0) .
\]
Then we have
\begin{eqnarray}
\| T \xi -T \eta \| & \leq & |\xi (0) -\eta (0) | + \sup _{0 \leq s \leq t} e^{- \lambda s}
\left| \int ^s _0 [b\big(\xi (r)\big) -b\big(\eta (r)\big)]\,dr \right| \nonumber \\
& \leq & |\xi (0) -\eta (0) | + \sup _{0 \leq s \leq t} e^{- \lambda s}
\kappa \int ^s _0 | \xi (r) -\eta (r) |\,dr \nonumber \\
& \leq & |\xi (0) -\eta (0) | + \sup _{0 \leq s \leq t} e^{- \lambda s}
\kappa \int ^s _0 e^{\lambda r} \| \xi -\eta \|\,dr \nonumber \\
&=& |\xi (0) -\eta (0) | + \alpha \| \xi-\eta \|, 
\end{eqnarray}
where $\alpha = \kappa / \lambda < 1$. For $x_0$ in $\Bbb{R} ^{\ell}$, let 
$\mathscr{X} _{x_0} =\{\xi \in \mathscr{X} : \xi (0) = x_0 \}$. Then $\mathscr{X} _{x_0}$ 
is a complete metric space and by (8.5), $T$ is a proper contraction on it. Therefore $T$ 
has a unique fixed point $x$ in $\mathscr{X} _{x_0}$. Since $t$ is arbitrary, there is a 
unique continuous function $x$ from $[0, \infty)$ to $\Bbb{R} ^{\ell}$ satisfying 
(8.2). Any solution of (8.2) is continuous, so there is a unique solution of (8.2).

Next we shall show that $P^t : C(\Bbb{R} ^{\ell}) \to 
C(\Bbb{R} ^{\ell})$. By (8.5) and induction on~$n$,
\begin{equation}
\| T^n \xi -T^n \eta \| \leq [1+\alpha + \ldots +\alpha ^{n-1}] \,|\xi (0)-\eta (0)| +
\alpha ^n \|\xi -\eta \|.
\end{equation}
If $x_0$ is in $\Bbb{R} ^{\ell}$, we shall also let $x_0$ denote the constant map 
$x_0 (s) =x_0$, and we shall let $x$ be the fixed point of $T$ with $x(0) =x_0$, so that
\[
x = \lim _{n \to \infty} T^n x_0,
\]
and similarly for $y_0$ in $\Bbb{R} ^{\ell}$. By (8.6), $\| x-y \| \leq \beta |x_0-y_0|$, 
where $\beta =1/(1-\alpha)$. Therefore, $|x(t) -y(t)| \leq e^{\lambda t} \beta |x_0-y_0|$. 
Now let $f$ be any Lipschitz function on $\Bbb{R} ^{\ell}$ with Lipschitz constant $K$. 
Then
\[
\left|f\big(x(t)\big)-f\big(y(t)\big)\right| \leq K e^{\lambda t} \beta |x_0-y_0|.
\]
Since this is true for each fixed $w$ path, the estimate remains true when we take 
expectations, so that
\[
|P^t f(x_0) -P^t f(y_0)| \leq K e^{\lambda t} \beta |x_0-y_0|.
\]
Therefore, if $f$ is a Lipschitz function in $C(\Bbb{R} ^{\ell})$ then $P^t f$ is a bounded 
continuous function. The Lipschitz functions are dense in $C(\Bbb{R} ^{\ell})$ and 
$P^t$ is a bounded linear operator. Consequently, if $f$ is in $C(\Bbb{R} ^{\ell})$ then 
$P^t f$ is a bounded continuous function. We still need to show that it vanishes at infinity. 
By uniqueness,
\[
x(t) = x(s) + \int ^t _s b\big(x(r)\big)\,dr +w(t)-w(s)
\]
for all $0 \leq s \leq t$, so that
\begin{eqnarray*}
|x(t)-x(s)| &\leq & \left| \int ^t _s [b\big(x(r)\big)-b\big(x(t)\big)]\,dr\right|
+(t-s)\left|b\big(x(t)\big)\right| \\
&&\qquad\qquad{}+|w(t)-w(s)| \\
& \leq & \kappa \int ^t _s |x(r)-x(t)|\,dr +t\left|b\big(x(t)\big)\right| +|w(t)-w(s)| \\
& \leq & \kappa \sup _{0 \leq r \leq t} |x(r)-x(t)| +t\left|b\big(x(t)\big)\right| 
+\sup _{0 \leq r \leq t} |w(t)-w(r)| .
\end{eqnarray*}
Since this is true for each $s$, $0 \leq s \leq t$,
\[
\sup _{0 \leq s \leq t} |x(t)-x(s)| \leq \gamma \big[\,t\left|b\big(x(t)\big)\right|
+\sup _{0 \leq s \leq t} |w(t)-w(s)| \,\big],
\]
where $\gamma = 1/(1-\kappa t)$, provided that $\kappa t <1$. In particular, if 
$\kappa t <1$ then
\begin{equation}
|x(t)-x_0| \leq \gamma \big[\,t\left|b\big(x(t)\big)\right| +\sup _{0 \leq s \leq t}|w(t)-w(s)| \,\big].
\end{equation}
Now let $f$ be in $C_{{\rm com}} (\Bbb{R} ^{\ell})$, let $\kappa t <1$, and let $\delta$ be 
the supremum of $|b(z_0)|$ for $z_0$ in the support of $f$. By (8.7), $f\big(x(t)\big)=0$ unless
\begin{equation}
\inf _{z_0 \in \operatorname{supp} f} |z_0 -x_0| \leq \gamma [\,t \delta +
\sup _{0 \leq s \leq t}|w(t)-w(s)| \,].
\end{equation}
But as $x_0$ tends to infinity, the probability that $w$ will satisfy (8.8) tends to~0. Since 
$f$ is bounded, this means that $\mathrm{E} f\big(x(t)\big)=P^t f(x_0)$ tends to 0 as $x_0$ 
tends to infinity. We have already seen that $P^t f$ is continuous, so $P^t f$ is in 
$C(\Bbb{R} ^{\ell})$. Since $C_{{\rm com}} (\Bbb{R} ^{\ell})$ is dense in 
$C(\Bbb{R} ^{\ell})$ and $P^t$ is a bounded linear operator, $P^t$ maps 
$C(\Bbb{R} ^{\ell})$ into itself, provided $\kappa t <1$. This restriction could have 
been avoided by introducing an exponential factor, but this is not necessary, as we shall 
show that the $P^t$ form a semigroup.

Let $0 \leq s \leq t$. The conditional distribution of $x(t)$, 
with $x(r)$ for all $0\le r\le s$ given, is a function of $x(s)$ alone,
since the equation
\[
x(t) = x(s) + \int ^t _s b\big(x(s')\big)\, ds' +w(t) -w(s),
\]
has a unique solution. Thus the $x$ process is a Markov process, and
\[
\mathrm{E} \{f\big(x(t)\big) \,|\, x(r), 0 \leq r \leq s \} = \mathrm{E} \{f\big(x(t)\big) \,|\, x(s) \} =P^{t-s} f\big(x(s)\big)
\]
for $f$ in $C(\Bbb{R} ^{\ell})$, $0 \leq s \leq t$. Therefore,
\begin{align*}
P^{t+s}f(x_0) & = \mathrm{E} f(x(t+s)\big) \\
& = \mathrm{E} \mathrm{E} \{ f\big(x(t+s)\big) \,|\, x(r), 0 \le r \le s \} \\
& = \mathrm{E} P^tf\big(x(s)\big) \\
& = P^s P^t f(x_0),
\end{align*}
so that $P^{t+s} = P^t P^s$. It is clear that
\[
\sup _{0 \leq f \leq 1} P^t f(x_0) =1
\]
for all $x_0$ and $t$.

It remains only to prove (8.4) for $f$ in $C^2 _{{\rm com}} (\Bbb{R} ^{\ell})$. 
(Since $C^2 _{{\rm com}} (\Bbb{R} ^{\ell})$ is dense in $C(\Bbb{R} ^{\ell})$ and the $P^t$ 
have norm one, this will imply that $P^t f \to f$ as $t \to 0$ for all $f$ in 
$C(\Bbb{R} ^{\ell})$, so that $P^t$ is a Markovian semigroup.)

Let $f$ be in $C^2 _{{\rm com}} (\Bbb{R} ^{\ell})$, and let $K$ be a compact 
set containing the support of $f$ in its interior. An argument entirely analogous to the 
derivation of (8.7), with the subtraction and addition of $b(x_0)$ instead 
of $b\big(x(t)\big)$, gives
\begin{equation}
|x(t)-x_0| \leq \gamma [\,t \,|b(x_0)| +\sup _{0 \leq s \leq t}|w(0)-w(s)| \,],
\end{equation}
provided $\kappa t <1$ (which we shall assume to be the case). Let $x_0$ be in the 
complement of $K$. Then $f(x_0)=0$ and $f\big(x(t)\big)$ is also 0 unless $\varepsilon 
\leq |x(t) -x_0|$, where $\varepsilon$ is the distance from the support of $f$ to the 
complement of $K$. But the probability that the right hand side of (8.9) will be bigger 
than $\varepsilon$ is ${\rm o}(t)$ (in fact, ${\rm o}(t^n)$ for all $n$) by familiar properties of the 
Wiener process. Since $f$ is bounded, this means that $P^t f(x_0)$ is uniformly ${\rm o}(t)$ 
for $x_0$ in the complement of $K$, so that
\[
\frac{P^t f(x_0)-f(x_0)}{t} \to b(x_0) \cdot \nabla f(x_0) + C f(x_0) =0
\]
uniformly for $x_0$ in the complement of $K$. Now let $x_0$ be in $K$. We have
\[
P^t f(x_0) = \mathrm{E} f\big(x(t)\big) = \mathrm{E} f\left(x_0 +\int ^t _0 b\big(x(s)\big)\, ds + w(t)-w(0)\right).
\]
Define $R(t)$ by
\begin{multline*}
f\left(x_0 +\int ^t _0 b\big(x(s)\big)\, ds + w(t)-w(0)\right) \\
=f(x_0) +tb(x_0) \cdot \nabla f(x_0) +[w(t)-w(0)] \cdot \nabla f(x_0) \\
+\frac 12 \sum _{i,j} [w^i(t) -w^i(0)][w^j(t)-w^j(0)] \frac{\partial ^2}{\partial 
x^i \partial x^j}f(x_0) +R(t).
\end{multline*}
Then
\[
\frac{P^t f(x_0)-f(x_0)}{t} = b(x_0) \cdot \nabla f(x_0) + C f(x_0) + \frac 1t \mathrm{E} R(t).
\]
By Taylor's formula,
\[
R(t) = {\rm o}(|w(t) -w(0)| ^2) +{\rm o}\left(\int ^t _0 \big[b\big(x(s)\big)-b(x_0)\big]\,ds\right).
\]
Since $\mathrm{E} (|w(t)-w(0)|^2) \leq \mathrm{const.} \; t$, we need only show that
\begin{equation}
\mathrm{E} \sup _{x_0 \in K} \frac 1t \int ^t _0 |b\big(x(s)\big)-b(x_0)|ds
\end{equation}
tends to 0. But (8.10) is less than
\[
\mathrm{E} \sup _{x_0 \in K} \frac 1t \kappa\int ^t _0 |x(s)-x_0|ds,
\]
which by (8.9) is less than
\begin{equation}
\mathrm{E} \sup _{x_0 \in K} \kappa \gamma [\,t\,|b(x_0)| + \sup _{0 \leq s \leq t} |w(0) -w(s)|].
\end{equation}
The integrand in (8.11) is integrable and decreases to 0 as $t \to 0$. ~~\textrm{QED}. \\

Theorem 8.1 can be generalized in various ways. The first paragraph of 
the theorem remains true if $b$ is a continuous function of $x$ and $t$ that satisfies a 
global Lipschitz condition in $x$ with a uniform Lipschitz constant for each compact 
$t$-interval. The second paragraph needs to be slightly modified as we no longer have 
a semigroup, but the proofs are the same. Doob [15, \S6, pp.\ 273--291], using K. It\^{o}'s 
stochastic integrals (see Chapter 11), has a much deeper generalization in which the matrix 
$c^{ij}$ depends on $x$ and $t$. The restriction that $b$ satisfy a global Lipschitz condition 
is necessary in general. For example, if the matrix $c^{ij}$ is 0 then we have a system of 
ordinary differential equations. However, if $C$ is elliptic (that is, if the matrix $c^{ij}$ 
is of positive type and non-singular) the smoothness conditions on $b$ can be greatly 
relaxed (cf.\ [20]).

We make the convention that
\[
dx(t) =b\big(x(t)\big)dt +dw(t)
\]
means that
\[
x(t) -x(s) = \int ^t_s b\big(x(r)\big)\,dr +w(t)-w(s)
\]
for all $t$ and $s$.
\\

%\begin{thm}
T{\ninerm HEOREM} 8.2 {\it
Let $A: \Bbb{R} ^{\ell} \to \Bbb{R} ^{\ell}$ be linear, let $w$ be a Wiener process on 
$\Bbb{R} ^{\ell}$ with infinitesimal generator {\rm (8.1)}, and let $f:[0, \infty) \to 
\Bbb{R} ^{\ell}$ be continuous. Then the solution of
\begin{equation}
dx(t) =Ax(t)dt +f(t)dt +dw(t), \quad x(0) =x_0,
\end{equation}
for $t \geq 0$ is
\begin{equation}
x(t) =e^{At} x_0 +\int ^t _0 e^{A(t-s)} f(s)\,ds + \int ^t _0 e^{A(t-s)} dw(s) .
\end{equation}
The $x(t)$ are Gaussian with mean
\begin{equation}
\mathrm{E} x(t) = e^{At} x_0 +\int ^t _0 e^{A(t-s)} f(s)\,ds
\end{equation}
and covariance $r(t,s) = \mathrm{E} \big(x(t)- \mathrm{E}x(t)\big)\big(x(s)-\mathrm{E}x(s)\big)$ given by
\begin{equation}
r(t,s)=
\begin{cases}
e^{A(t-s)} \int ^s _0 e^{Ar} 2 c e^{A^\mathrm{T} r} dr, & t \geq s \\
\int ^t _0 e^{Ar} 2 c e^{A^\mathrm{T} r} dre^{A(s-t)}, & t \leq s .
\end{cases}
\end{equation}
}
%\end{thm}

The latter integral in (8.13) is a Wiener integral (as in Chapter 7). In (8.15), 
$A^\mathrm{T}$ denotes the transpose of $A$ and $c$ is the matrix with entries $c^{ij}$ 
occurring in (8.1). \\

{\it Proof.} Define $x(t)$ by (8.13). Integrate the last term in 
(8.13) by parts, obtaining
\begin{eqnarray*}
\int ^t _0 e^{A(t-s)} dw(s) &=& \int ^t _0 Ae^{A(t-s)} w(s)\,ds+e^{A(t-s)} w(s) \big\vert ^{s=t} _{s=0} \\
&=& \int ^t _0 Ae^{A(t-s)} w(s)\,ds+w(t)-e^{At} w(0) . 
\end{eqnarray*}
It follows that $x(t)-w(t)$ is differentiable, and has derivative $Ax(t) +f(t)$. This 
proves that (8.12) holds.

The $x(t)$ are clearly Gaussian with the mean (8.14). Suppose that 
$t \geq s$. Then the covariance is given by
\begin{eqnarray*}
\lefteqn{\mathrm{E} x_i (t) x_j (s) -\mathrm{E} x_i(t) \mathrm{E} x_j (s)} \\
&=& \mathrm{E} \int ^t _0 \sum _k \big(e^{A(t-t_1)}\big)_{ik} \,dw_k (t_1) 
\int ^s _0 \sum _h \big(e^{A(s-s_1)}\big)_{jh} \,dw_h (s_1) \\
&=& \int ^s _0 \sum _{k,h} \big(e^{A(t-r)}\big)_{ik} 2c_{kh} \big(e^{A(s-r)}\big)_{jh} \,dr \\
&=& \int ^s _0 \left(e^{A(t-r)}2ce^{A^\mathrm{T}(s-r)}\right)_{ij} \,dr \\
&=& \left(e^{A(t-s)} \int ^s _0 e^{Ar} 2ce^{A^\mathrm{T}r}dr\right)_{ij} .
\end{eqnarray*}
The case $t \leq s$ is analogous. ~~\textrm{QED}.

%~ \\
\begin{center}
\textrm{\bf Reference}
\end{center}
[20]. Edward Nelson,\emph{ Les \'{e}coulements incompressibles d'\'{e}nergie finie}, 
Colloques internationaux du Centre national de la recherche scientifique N$^{ {\rm o}}$ 117, 
``Les \'{e}quations aux d\'{e}riv\'{e}es partielles'', \'Editions du C.N.R.S., Paris, 1962. 
(The last statement in section II is incorrect.)

%\newpage
%\thispagestyle{empty}
%~

\newpage

\fancyhead[CO]{{\footnotesize THE ORNSTEIN-UHLENBECK THEORY OF BROWNIAN MOTION}}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{The Ornstein-Uhlenbeck theory of Brownian motion}

\hspace{6mm}The theory of Brownian motion developed by Einstein and Smoluchowski, although in agreement 
with experiment, was clearly a highly idealized treatment. The theory was far removed from the 
Newtonian mechanics of particles. Langevin initiated a train of thought that, in 1930, culminated 
in a new theory of Brownian motion by L. S. Ornstein and G. E. Uhlenbeck [22]. For ordinary Brownian 
motion (e.g., carmine particles in water) the predictions of the Ornstein-Uhlenbeck theory are 
numerically indistinguishable from those of the Einstein-Smoluchowski theory. However, the 
Ornstein-Uhlenbeck theory is a truly dynamical theory and represents great progress in the 
understanding of Brownian motion. Also, as we shall see later (Chapter 10), there is a Brownian motion 
where the Einstein-Smoluchowski theory breaks down completely and the Ornstein-Uhlenbeck 
theory is successful. 

The program of reducing Brownian motion to Newtonian particle mechanics is still incomplete. 
The problem, or one formulation of it, is to deduce each of the following theories from 
the one below it:
\begin{center}
\begin{tabular}{lll}
Einstein&-&Smoluchowski \\
Ornstein&-&Uhlenbeck \\
Maxwell&-&Boltzmann \\
Hamilton&-&Jacobi.
\end{tabular}
\end{center}
We shall consider the first of these reductions in detail later (Chapter 10). Now we shall describe the 
Ornstein-Uhlenbeck theory for a free particle and compare it with Einstein's theory.

We let $x(t)$ denote the position of a Brownian particle at time $t$ and assume that the velocity 
$dx/dt = v$ exists and satisfies the Langevin equation
\begin{equation}
dv(t) =-\beta v(t) dt + dB(t) .
\end{equation}
Here $B$ is a Wiener process (with variance parameter to be determined later) and $\beta$ is 
a constant with the dimensions of frequency (inverse time). Let $m$ be the mass of the particle, 
so that we can write
\[
m \frac{d^2 x}{d t^2} =-m \beta v + m \frac{d B}{dt} .
\]
This is merely formal since $B$ is not differentiable. Thus (using Newton's law $F=ma$) we are 
considering the force on a free Brownian particle as made up of two parts, a frictional force 
$F_0 = -m \beta v$ with friction coefficient $m \beta$ and a fluctuating force $F_1 =mdB/dt$ 
which is (formally) a Gaussian stationary process with correlation function of the form a constant 
times $\delta$, where the constant will be determined later.

If $v(0) = v_0$ and $x(0) = x_0$, the solution of the initial value problem is, by Theorem 8.2,
\begin{equation}
\begin{split}
v(t) &= e^{- \beta t} v_0 +e^{- \beta t} \int ^t _0 e^{\beta s} dB(s), \\
x(t) &= x_0 + \int ^t _0 v(s)\, ds.
\end{split}
\end{equation}
For a free particle there is no loss of generality in considering only the case of one-dimensional 
motion. Let $\sigma ^2$ be the variance parameter of $B$ (infinitesimal generator $\frac 12 
\sigma ^2 d^2 /dv^2$, $\mathrm{E} dB(t)^2 = \sigma ^2 dt$). The velocity $v(t)$ is 
Gaussian with mean
\[
e^{- \beta t} v_0,
\]
by (9.2). To compute the covariance, let $t \ge s$. Then 
\begin{align*}
\mathrm{E} \Bigg(e^{- \beta t}\int ^t _0 e^{\beta t_1} & dB(t_1) e^{- \beta s} 
\int ^s _0 e^{\beta s_1} dB(s_1)\Bigg) \\
& = e^{- \beta (t+s) } \int ^s _0 e^{2 \beta r} \sigma ^2 dr \\
& = e^{- \beta (t+s) } \sigma ^2 \frac{e^{2 \beta s}-1}{2 \beta}.
\end{align*}
For $t=s$ this is 
\[
\frac{\sigma ^2}{2 \beta} (1- e^{-2 \beta t}) .
\]
Thus, no matter what $v_0$ is, the limiting distribution of $v(t)$ as $t \to \infty$ is Gaussian 
with mean 0 and variance $\sigma ^2 /2 \beta$. Now the law of equipartition of energy in 
statistical mechanics says that the mean energy of the particle (in equilibrium) per degree 
of freedom should be $\frac 12 kT$. Therefore we set 
\[
\frac 12 m \frac{\sigma ^2 }{2 \beta} = \frac 12 kT .
\]
That is, recalling the previous notation $D=kT/m \beta$, we adopt the notation
\[
\sigma ^2 = 2 \frac{\beta kT}{m} =2 \beta ^2 D
\]
for the variance parameter of $B$.

We summarize in the following theorem.
\\

%\begin{thm}
T{\ninerm HEOREM} 9.1 {\it
Let $D$ and $\beta$ be strictly positive constants and let $B$ be the Wiener process on $\Bbb{R}$ 
with variance parameter $2 \beta ^2 D$. The solution of 
\[
dv(t) =-\beta v(t) dt + dB(t); \quad v(0) = v_0
\]
for $t>0$ is 
\[
v(t) = e^{- \beta t} v_0 + \int ^t _0 e^{- \beta (t-s)} dB(s).
\]
The random variables $v(t)$ are Gaussian with mean
\[
m(t) = e^{- \beta t} v_0
\]
and covariance
\[
r(t,s) = \beta D\left(e^{- \beta |t-s|}-e^{- \beta (t+s)}\right).
\]
The $v(t)$ are the random variables of the Markov process on $\Bbb{R}$ with infinitesimal generator 
\[
- \beta v \frac{d}{dv}+ \beta ^2 D \frac{d^2}{dv^2}
\]
with domain including $C^2 _{{\rm com}} (\Bbb{R})$, with initial measure $\delta _{v_0}$. The kernel 
of the corresponding semigroup operator $P^t$ is given by
\[
p^t (v_0,dv) = [2 \pi \beta D(1-e^{-2 \beta t})]^{- \frac 12} 
\exp \left[- \frac{(v-e^{- \beta t}v_0)^2}{2 \beta D (1-e^{-2 \beta t})}\right] dv .
\]
The Gaussian measure $\mu$ with mean 0 and variance $\beta D$ is invariant, $P^{t*} \mu = \mu$, 
and $\mu$ is the limiting distribution of $v(t)$ as $t \to \infty$.
}
%\end{thm}
\\

The process $v$ is called the Ornstein-Uhlenbeck velocity process with diffusion coefficient $D$ and 
relaxation time $\beta ^{-1}$, and the corresponding position process~$x$
$\big($given by (9.2)$\big)$ is called the Ornstein-Uhlenbeck process.
\\

%\begin{thm}
T{\ninerm HEOREM} 9.2 {\it
Let the $v(t)$ be as in Theorem {\rm 9.1}, and let 
\[
x(t) = x_0 + \int ^t _0 v(s)\, ds.
\]
Then the $x(t)$ are Gaussian with mean
\[
\tilde{m} (t) = x_0 + \frac{1-e^{- \beta t}}{\beta}v_0
\]
and covariance
\[
\tilde{r} (t,s) = 2D \min (t,s) + \frac{D}{\beta} \left(-2+2e^{- \beta t}+2e^{- \beta s}-e^{- \beta |t-s|}
-e^{- \beta (t+s)}\right).
\]
}
%\end{thm}

{\it Proof.} This follows from Theorem 9.1 by integration,
\begin{align*}
\tilde{m} (t) & = x_0 + \int ^t _0 m(s)\, ds, \\
\tilde{r} (t,s) & = \int ^t _0 dt_1 \int ^s _0 ds_1 \,r(t_1,s_1) .
\end{align*}
The second integration is tedious but straightforward. ~~\textrm{QED}. \\

In particular, the variance of $x(t)$ is 
\[
2Dt + \frac{D}{\beta} (-3 +4e^{- \beta t}-e^{- 2 \beta t}).
\]
The variance in Einstein's theory is $2Dt$. By elementary calculus, the absolute value of the 
difference of the two variances is less than $3D \beta ^{-1}$. In the typical case of 
$\beta ^{-1} = 10 ^{-8} \sec.$, $t = \frac 12 \sec.$, we make a proportional error of less than 
$3 \times 10^{-8}$ by adopting Einstein's value for the variance. The following theorem shows 
that the Einstein theory is a good approximation to the Ornstein-Uhlenbeck theory for a free particle.
\\

%\begin{thm}
T{\ninerm HEOREM} 9.3 {\it
Let $0 = t_0 < t_1 < \ldots < t_n $, and let 
\[
\Delta t= \min _{1 \le i \le n} t_i-t_{i-1} .
\]
Let $f(x_1, \ldots, x_n)$ be the probability density function for $x(t_1), \ldots, x(t_n)$, where $x$ 
is the Ornstein-Uhlenbeck process with $x(0) = x_0$, $v(0)=v_0$, diffusion coefficient $D$ and 
relaxation time $\beta ^{-1}$. Let $g(x_1, \ldots, x_n)$ be the probability density function for 
$w(t_1), \ldots, w(t_n)$, where $w$ is the Wiener process with $w(0) = x_0$ and diffusion 
coefficient $D$.

Let $\varepsilon >0$. There exist $N_1$ depending only on $\varepsilon$ and $n$ and $N_2$ 
depending only on $\varepsilon$ such that if 
\begin{equation}
\Delta t \ge N_1 \beta ^{-1},
\end{equation}
\begin{equation}
t_1 \ge N_2 \frac{v_0 ^2}{2D \beta ^2},
\end{equation}
then
\begin{equation}
\int _{\Bbb{R} ^n} |f(x_1, \ldots, x_n)-g(x_1, \ldots, x_n)|\,dx_1 \ldots dx_n \le \varepsilon .
\end{equation}
}
%\end{thm}

{\it Proof.} Assume, as one may without loss of generality, that $x_0=0$. Consider the 
non-singular linear transformation 
$$(x_1, \ldots, x_n) \mapsto (\tilde{x}_1, \ldots, \tilde{x}_n)$$
on 
$\Bbb{R}^n$ given by
\begin{equation}
\tilde{x}_i = [2D(t_i-t_{i-1})]^{-\frac 12} (x_i-x_{i-1})
\end{equation}
for $i=1, \ldots, n$. The random variables $\tilde{w} (t_i)$ obtained when this transformation 
is applied to the $w(t_i)$ are orthonormal since $\mathrm{E} w(t_i)w(t_j)=2D \min (t_i,t_j)$. 
Thus $\tilde{g}$, the probability density function of the $\tilde{w} (t_i)$, is the unit Gaussian 
function on $\Bbb{R}^n$. Let $\tilde{f}$ be the probability density function of the $\tilde{x} (t_i)$, 
where the $\tilde{x} (t_i)$ are obtained by applying the linear transformation (9.6) to the $x(t_i)$. 
The left hand side of (9.5) is unchanged when we replace $f$ by $\tilde{f}$ and $g$ by $\tilde{g}$, since 
the total variation norm of a measure is unchanged under a one-to-one measurability-preserving 
map such as (9.6).

We use the notation $\mathrm{Cov}$ for the covariance of two random variables, $\mathrm{Cov}\, 
xy = \mathrm{E} xy -\mathrm{E} x \mathrm{E} y$. By Theorem 9.2 and the remark following it,
\[
\mathrm{Cov}\, x(t_i) x(t_j) = \mathrm{Cov}\,w(t_i) w(t_j) + \varepsilon _{ij},
\]
where $|\varepsilon _{ij} | \le 3D \beta ^{-1}$. By (9.6),
\[
\mathrm{Cov}\, \tilde{x} (t_i) \tilde{x} (t_j) = \delta _{ij} + \varepsilon _{ij}',
\]
where $|\varepsilon _{ij}' | \le 4 \cdot 3D \beta ^{-1}/2D \Delta t \le 6/N_1$ if (9.3) holds. 
Again by Theorem 9.2, the mean of $\tilde{x} (t_1)$ is, in absolute value, smaller than
\[
|v_0| /\beta [2Dt_1]^{\frac 12} \le N_2 ^{- \frac 12}
\]
if (9.4) holds. The mean of $\tilde{x} (t_i)$ for $i >1$ is, in absolute value, smaller than 
\[
(e^{- \beta t_{i-1}}-e^{- \beta t_i}) |v_0|/\beta[2D(t_i -t_{i-1})]^{\frac 12} .
\]
Since the first factor is smaller than 1, the square of this is smaller than
\[
\frac{e^{- \beta t_{i-1}}-e^{- \beta t_i}}{t_i-t_{i-1}} \frac{v_0 ^2}{2D \beta ^2} 
\le \frac{\beta t_1}{N_2}e^{- \beta t_1} \le \frac{N_1 e^{-N_1}}{N_2}
\]
if (9.3) and (9.4) hold with $N_1 \ge 1$. Therefore, if we choose $N_1$ and $N_2$ 
large enough, the mean and covariance of $\tilde{f}$ are arbitrarily close to 0 and $\delta _{ij}$, 
respectively, which concludes the proof. ~~\textrm{QED}. \\

Chandrasekhar omits the condition (9.4) in his discussion [21, equations (171) through (174)], 
but his reasoning is circular. Clearly, if $v_0$ is enormous then $t_1$ must be suitable large 
before the Wiener process is a good approximation. The condition (9.3) is usually written 
$\Delta t \gg \beta ^{-1}$ \hbox{($\Delta t$ much larger than $\beta ^{-1}$)}. If $v_0$ is a typical 
velocity---i.e., if $|v_0|$ is not much larger than the standard deviation 
$(kT/m)^{\frac 12} = (D \beta)^{\frac 12}$ 
of the Maxwellian velocity distribution---then the condition (9.4), $t_1 \gg v_0 ^2 /2D \beta ^2$, 
is no additional restriction if $\Delta t \gg \beta ^{-1}$.

There is another, and quite weak, formulation of the fact that the Wiener process is a good 
approximation to the Ornstein-Uhlenbeck process for a free particle in the limit of very large 
$\beta$ (very short relaxation time) but $D$ of reasonable size. \\

D{\ninerm EFINITION.} Let $x_{\alpha}$, $x$ be real stochastic processes indexed by the same index 
set $T$ but not necessarily defined on a common probability space. We say that $x_{\alpha}$ 
\emph{converges to} $x$ \emph{in distribution} in case for each $t_1, \ldots, t_n$ in $T$, the 
distribution of $x_{\alpha} (t_1), \ldots,x_{\alpha} (t_n)$ converges (in the weak-$^{*}$ topology 
of measures on $\Bbb{R}^n$, as $\alpha$ ranges over a directed set) to the distribution of 
$x(t_1), \dots,x(t_n)$. \\

It is easy to see that if we represent all of the processes in the usual way [25] on $\Omega =
\dot{\Bbb{R}}^I$, this is the same as saying that $\mathrm{Pr} _{\alpha}$ converges to 
$\mathrm{Pr}$ in the weak-$*$ topology of regular Borel measures on $\Omega$, where 
$\mathrm{Pr} _{\alpha}$ is the regular Borel measure associated with $x_{\alpha}$ and 
$\mathrm{Pr}$.

The following two theorems are trivial.
\\

%\begin{thm}
T{\ninerm HEOREM} 9.4 {\it
Let $x_{\alpha}$, $x$ be Gaussian stochastic processes with means $m_{\alpha}$, $m$ and 
covariances $r_{\alpha}$, $r$. Then $x_{\alpha}$ converges to $x$ in distribution if and only if 
$r_{\alpha} \to r$ and $m_{\alpha} \to m$ pointwise (on $T$ and $T \times T$ respectively, 
where $T$ is the common index set of the processes).
}
\\
%\end{thm}

%\begin{thm}
T{\ninerm HEOREM} 9.5 {\it
Let $\beta$ and $\sigma ^2$ vary in such a way that $\beta \to \infty$ and $D= \sigma ^2 /2 \beta ^2$ 
remains constant. Then for all $v_0$ the Ornstein-Uhlenbeck process with initial conditions 
$x(0) = x_0$, $v(0)=v_0$, diffusion coefficient $D$, and relaxation time $\beta ^{-1}$ converges 
in distribution to the Wiener process starting at $x_0$ with diffusion coefficient $D$.
}
\\
%\end{thm}

%~ \\[1.5cm]
\begin{center}
\textrm{\bf References}
\end{center}

The best account of the Ornstein-Uhlenbeck theory and related matters is:
\smallskip

\noindent [21]. S. Chandrasekhar, \emph{Stochastic problems in physics and astronomy}, 
Reviews of Modern Physics \textrm{15} (1943), 1--89.

See also:
\smallskip

\noindent [22]. G. E. Uhlenbeck and L. S. Ornstein, \emph{On the theory of 
Brownian motion}, Physical Review \textrm{36} (1930), 823--841.
\smallskip

\noindent [23]. Ming Chen Wang and G. E. Uhlenbeck, \emph{On the theory of 
Brownian motion} II, Reviews of Modern Physics \textrm{17} (1945), 323--342. 
\smallskip

The first mathematically rigorous treatment, and additionally the 
source of great conceptual and computational simplifications, was:
\smallskip

\noindent [24]. J. L. Doob, \emph{The Brownian movement and stochastic equations}, 
Annals of Mathematics \textrm{43} (1942), 351--369.
\smallskip

All four of these articles are reprinted in the Dover paperback 
``Selected Papers on Noise and Stochastic Processes'', edited by Nelson Wax.

\smallskip

\noindent [24]. E. Nelson, \emph{Regular probability measures on function space}, 
Annals of Mathematics \textrm{69} (1959), 630--643.

\newpage

\fancyhead[CO]{BROWNIAN MOTION IN A FORCE FIELD}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Brownian motion in a force field}

\hspace{6mm}We continue the discussion of the Ornstein-Uhlenbeck theory. Suppose we have a Brownian 
particle in an external field of force given by $K(x,t)$ in units of force per unit mass (acceleration). 
Then the Langevin equations of the Ornstein-Uhlenbeck theory become
\begin{equation}
\begin{split}
dx(t) & = v(t)dt \\
dv(t) & = K\big(x(t),t\big)dt-\beta v(t)dt + dB(t),
\end{split}
\end{equation}
where $B$ is a Wiener process with variance parameter $2 \beta ^2 D$. This is of the form 
considered in Theorem 8.1:
\[
d \binom{x(t)}{v(t)} = \binom{v(t)}{K\big(x(t),t\big)-\beta v(t)} dt + d \binom{0}{B(t)} .
\]
Notice that we can no longer consider the velocity process, or a component of it, by itself.

For a free particle ($K=0$) we have seen that the Wiener process, which is a Markov process 
on coordinate space ($x$-space) is a good approximation, except for very small time intervals, 
to the Ornstein-Uhlenbeck process, which is a Markov process on phase space ($x,v$-space). 
Similarly, when an external force is present, there is a Markov process on coordinate space, 
discovered by Smoluchowski, which under certain circumstances is a good approximation to 
the position $x(t)$ of the Ornstein-Uhlenbeck process.

Suppose, to begin with, that $K$ is a constant. The force on a particle of mass $m$ is $Km$ 
and the friction coefficient is $m \beta$, so the particle should acquire the limiting velocity 
$Km / m \beta = K/ \beta$. That is, for times large compared to the relaxation time $\beta ^{-1}$ 
the velocity should be approximately $K / \beta$. If we include the random fluctuations due to 
Brownian motion, this suggests the equation
\[
dx(t) = \frac{K}{\beta} dt + dw(t)
\]
where $w$ is the Wiener process with diffusion coefficient $D = kT/m \beta$. $\big($If there were no 
diffusion we would have, approximately for $t \gg \beta ^{-1}$, $dx(t)=(K/ \beta)dt$, and if there 
were no force we would have $dx(t)=dw(t)\big)$. If now $K$ depends on $x$ and $t$, but varies so slowly 
that it is approximately constant along trajectories for times of the order $ \beta ^{-1}$, we write
\[
dx(t) = \frac{K\big(x(t),t\big)}{\beta} dt + dw(t) .
\]
This is the basic equation of the Smoluchowski theory; cf.\ Chandrasek\-har's discussion [21].

We shall begin by discussing the simplest case, when $K$ is linear and independent of $t$. 
Consider the one-dimensional harmonic oscillator with circular frequency $\omega$. 
The Langevin equation in the Ornstein-Uhlenbeck theory is then
\begin{align*}
dx(t) & = v(t)dt \\
dv(t) & =-\omega ^2 x(t)dt-\beta v(t)dt + dB(t)
\end{align*}
or
\[
d \binom{x}{v} = \begin{pmatrix}
 0 & 1 \\
 -\omega ^2 &-\beta
 \end{pmatrix}
\binom{x}{v} dt + d \binom{0}{B},
\]
where, as before, $B$ is a Wiener process with variance parameter 
$\sigma ^2 = 2 \beta kT/m=2 \beta ^2 D$.

The characteristic equation of the matrix
\[
A = \begin{pmatrix}
 0 & 1 \\
 -\omega ^2 &-\beta
 \end{pmatrix}
\]
is $\mu ^2 + \beta \mu + \omega ^2 = 0$, with the eigenvalues
\[
\mu _1 =-\frac 12 \beta + \sqrt{\frac 14 \beta ^2-\omega ^2}, \qquad 
\mu _2 =-\frac 12 \beta-\sqrt{\frac 14 \beta ^2-\omega ^2}.
\]
As in the elementary theory of the harmonic oscillator without Brownian motion, 
we distinguish three cases: \\
\begin{center}
\begin{tabular}{ll}
overdamped & $\beta > 2 \omega$, \\
critically damped & $\beta = 2 \omega$, \\
underdamped & $\beta < 2 \omega$.
\end{tabular}
\end{center}
Except in the critically damped case, the matrix $\exp (tA)$ is
\[
e^{tA} = \frac{1}{\mu _2-\mu _1} 
\begin{pmatrix}
 \mu _2 e^{\mu _1 t}-\mu _1 e^{\mu _2 t} &-e^{\mu _1 t} + e^{\mu _2 t} \\
 \mu _2 \mu _1 e^{\mu _1 t}-\mu _2 \mu _1 e^{\mu _2 t} & 
 -\mu _1 e^{\mu _1 t} + \mu _2 e^{\mu _2 t}
\end{pmatrix} .
\]
(We derive this as follows. Each matrix entry must be a linear combination of $\exp ( \mu _1 t)$ 
and $\exp ( \mu _2 t)$. The coefficients are determined by the requirements that $\exp (tA)$ and 
$d\exp (tA) /dt$ are 1 and $A$ respectively when $t=0$.)

We let $x(0) = x_0$, $v(0) = v_0$. Then the mean of the process is
\[
e^{tA} \binom{x_0}{v_0} .
\]
The covariance matrix of the Wiener process $\binom{0}{B}$ is
\[
2c = \begin{pmatrix}
 0 & 0 \\
 0 & 2 \beta ^2 D
 \end{pmatrix} .
\]
The covariance matrix of the $x, v$ process can be determined from Theorem 8.2, but the formulas 
are complicated and not very illuminating. The covariance for equal times are listed by 
Chandrasekhar [21, original page 30].

The Smoluchowski approximation is
\[
dx(t) =-\frac{\omega ^2 }{\beta} x(t) dt + dw(t),
\]
where $w$ is a Wiener process with diffusion coefficient $D$. This has the same form as the 
Ornstein-Uhlenbeck velocity process for a free particle. According to the intuitive argument 
leading to the Smoluchowski equation, it should be a good approximation for time intervals 
large compared to the relaxation time ($\Delta t \gg \beta ^{-1}$) when the force is slowly 
varying ($\beta \gg 2 \omega$; i.e., the highly overdamped case).

The Brownian motion of a harmonically bound particle has been investigated experimentally by 
Gerlach and Lehrer and by Kappler [26]. The particle is a very small mirror suspended in a gas by 
a thin quartz fiber. The mirror can rotate but the torsion of the fiber supplies a linear restoring 
force. Bombardment of the mirror by the molecules of the gas causes a Brownian motion of the 
mirror. The Brownian motion is one-dimensional, being described by the angle that the mirror 
makes with its equilibrium position. (This angle, which is very small, can be measured accurately 
by shining a light on the mirror and measuring the position of the reflected spot a large distance 
away.) At atmospheric pressure the motion is highly overdamped, but at sufficiently low pressures 
the underdamped case can be observed, too. The Ornstein-Uhlenbeck theory gives for the invariant 
measure (limiting distribution as $t \to\infty$) $\mathrm{E} x^2 = kT/m \omega ^2$ and 
$\mathrm{E} v^2 = kT / m$. That is, the expected value of the kinetic energy $\frac 12 mv^2$ 
in equilibrium is $\frac 12 kT$, in accordance with the equipartition law of statistical mechanics. 
These values are independent of $\beta$, and the constancy of $\mathrm{E}x^2$ as the pressure varies 
was observed experimentally. However, the appearance of the trajectories varies tremendously.

Consider Fig.\ 5a on p.\ 243 of Kappler [26], which is the same as Fig.~5b on p.\ 169 of Barnes and 
Silverman [11, \S3]. This is a record of the motion in the highly overdamped case. Locally the 
graph looks very much like the Wiener process, extremely rough. However, the graph never rises 
very far above or sinks very far below a median position, and there is a general tendency to return 
to the median position. If we reverse the direction if time, the graph looks very much the same. 
This process is a Markov process---there is no memory of previous positions. A graph of the velocity 
in the Ornstein-Uhlenbeck process for a free particle would look the same.

Now consider Fig.\ 6a on p.\ 244 of Kappler (Fig.\ 5c on p.\ 169 of Barnes and Silverman). This is 
a record of the motion in the underdamped case. The curve looks smooth and more or less sinusoidal. 
This is clearly not the graph of a Markov process, as there is an evident distinction between the 
upswings and downswings of the curve. Consequently, the Smoluchowski approximation is 
completely invalid in this case. When Barnes and Silverman reproduced the graph, it got turned 
over, reversing the direction of time. However, the over-all appearance of the curves is 
very much the same and in fact this stochastic process is invariant under time reversal. 
Are there any beats in this graph, and should there be?

Fig.\ 4b on p.\ 242 of Kappler (fig.\ 5a on p.\ 169 of Barnes and Silverman) represents an 
intermediate case.

$$\vbox{\pdfimage{2a.png}}$$
$$\hbox{Figure 2a}$$

$$\vbox{\pdfimage{2b.png}}$$
$$\hbox{Figure 2b}$$

$$\vbox{\pdfimage{2c.png}}$$
$$\hbox{Figure 2c}$$
\bigskip

We illustrate crudely the two cases (Fig.\ 2). Fig.\ 2a is the highly overdamped case, a Markov 
process. Fig.\ 2b is the underdamped case, not a Markov process. Fig.\ 2c illustrates a case 
that does not occur. (The only repository for memory is in the velocity, so over-all sinusoidal 
behavior implies local smoothness of the curve.)

One has the feeling with some of Kappler's curves that one can occasionally see where an 
exceptionally energetic gas molecule gave the mirror a kick. This is not true. Even at the lowest 
pressure used, an enormous number of collisions takes place per period, and the irregularities in the 
curves are due to chance fluctuations in the sum of enormous numbers of individually negligible events.

It is not correct to think simply that the jiggles in a Brownian trajectory are due to kicks from 
molecules. Brownian motion is unbelievably gentle. Each collision has an entirely negligible effect 
on the position of the Brownian particle, and it is only fluctuations in the accumulation of an 
enormous number of very slight changes in the particle's velocity that give the trajectory its 
irregular appearance.

The experimental results lend credence to the statement that the Smoluchowski approximation is 
valid when the friction is large ($\beta$ large). A theoretical proof does not seem to be in the 
literature. Ornstein and Uhlenbeck [22] show only that if a harmonically bound particle starts at 
$x_0$ at time 0 with a Maxwellian-distributed velocity, the mean and variance of $x(t)$ are 
approximately the mean and variance of the Smoluchowski theory provided $\beta \gg 2 \omega$ 
and $\Delta t \gg \beta ^{-1}$. We shall examine the Smoluchowski approximation in the case of a general
external force, and prove a result that says that it is in a very strong sense the limiting case of 
the Ornstein-Uhlenbeck theory for large friction.

Consider the equations (10.1) of the Ornstein-Uhlenbeck theory. Let $w$ be a Wiener process with 
diffusion coefficient $D$ (variance parameter~$2D$) as in the Einstein-Smoluchowski theory. Then if 
we set $B = \beta w$ the process $B$ has the correct variance parameter $2 \beta ^2 D$ for the 
Ornstein-Uhlenbeck theory. The idea of the Smoluchowski approximation is that the relaxation time 
$\beta ^{-1}$ is negligibly small but that the diffusion coefficient $D=kT/m \beta$ and the velocity 
$K/ \beta$ are of significant size. Let us therefore define $b(x,t)$ (having the dimensions 
of a velocity) by
\[
b(x,t) = \frac{K(x,t)}{\beta}
\]
and study the solution of (10.1) as $\beta \to \infty$ with $b$ and $D$ fixed. 
The equations (10.1) become
\begin{align*}
dx(t) & = v(t) dt \\
dv(t) & =-\beta v(t) dt + \beta b\big(x(t),t\big) dt + \beta dw(t) .
\end{align*}
Let $x(t) = x(\beta, t)$ be the solution of these equations with $x(0)=x_0$, $v(0) = v_0$. We will 
show that as $\beta \to \infty$, $x(t)$ converges to the solution $y(t)$ of the Smoluchowski equation
\[
dy(t) = -b\big(y(t),t\big) dt +dw(t)
\]
with $y(0)= x_0$. For simplicity, we treat the case that $b$ is independent of the time, although 
the theorem and its proof remain valid for the case that $b$ is continuous and, for $t$ in compact 
sets, satisfies a uniform Lipschitz condition in $x$.
\\

%\begin{thm}
T{\ninerm HEOREM} 10.1 {\it
Let $b: \Bbb{R}^{\ell} \to \Bbb{R} ^{\ell}$ satisfy a global Lipschitz condition and let $w$ be a 
Wiener process on $\Bbb{R} ^{\ell}$. Let $x, v$ be the solution of the coupled equations
\begin{equation}
dx(t)  = v(t) dt; \quad\hbox{$x(0)=x_0$,\phantom{xxxxxxxxxxxxxxxxxxxxxxxxx..}}  
\end{equation}
\vskip-30pt
\begin{equation}
dv(t)  =-\beta v(t) dt + \beta b\big(x(t),t\big)dt + \beta dw(t);  \quad\hbox{$v(0)=v_0$.}
\end{equation}
Let $y$ be the solution of
\begin{equation}
dy(t) = b\big(y(t)\big) dt + dw(t); \qquad y(0) = x_0.
\end{equation}
For all $v_0$, with probability one
\[
\lim _{\beta \to \infty} x(t) = y(t),
\]
uniformly for $t$ in compact subintervals of $[ 0, \infty)$.
}
\\
%\end{thm}

{\it Proof.} Let $\kappa$ be the Lipschitz constant of $b$, so that $|b(x_1)-b(x_2)| \le \kappa 
|x_1-x_2| $ for all $x_1$, $x_2$ in $\Bbb{R} ^{\ell}$. Let
\[
t_n = n \frac{1}{2 \kappa}
\]
for $n=0,1,2, \ldots$. Consider the equations on $[t_n,t_{n+1}]$. By (10.2),
\begin{equation}
x(t) = x(t_n) + \int ^t _{t_n} v(s)\,ds,
\end{equation}
and by (10.3),
\begin{equation}
v(t) = v(t_n)-\beta \int ^t _{t_n} v(s)\,ds + \beta \int ^t _{t_n} b\big(x(s)\big)\,ds + \beta [w(t)-w(t_n)],
\end{equation}
or equivalently,
\begin{equation}
\int ^t _{t_n} v(s)\,ds = \frac{v(t_n)}{\beta}-\frac{v(t)}{\beta} +\int ^t _{t_n} b\big(x(s)\big)\,ds +w(t)-w(t_n) .
\end{equation}
By (10.5) and (10.7),
\begin{equation}
x(t) = x(t_n) + \frac{v(t_n)}{\beta}-\frac{v(t)}{\beta} +\int ^t _{t_n} b\big(x(s)\big)\,ds +w(t)-w(t_n) .
\end{equation}
By (10.4),
\begin{equation}
y(t) = y(t_n) + \int ^t _{t_n} b\big(y(s)\big)\,ds +w(t)-w(t_n),
\end{equation}
so that by (10.8) and (10.9),
\begin{equation}
\begin{split}
x(t)-y(t) = x(t_n) &-y(t_n) + \frac{v(t_n)}{\beta}-\frac{v(t)}{\beta} \\
& + \int ^t _{t_n} [b\big(x(s)\big) -b\big(y(s)\big)]ds
\end{split}
\end{equation}
The integral in (10.10) is bounded in absolute value by
\[
(t-t_n) \kappa \sup _{t_n \le s \le t_{n+1}} |x(s) -y(s)|,
\]
and $(t-t_n) \kappa \le \frac 12$ for $t_n \le t \le t_{n+1}$, so that
\begin{equation}
\begin{split}
|x(t)-y(t)| \le |x(t_n) &-y(t_n)| + 2 \sup _{t_n \le s \le t_{n+1}}\left|\frac{v(s)}{\beta}\right| \\
& + \frac 12 \sup _{t_n \le s \le t_{n+1}} |x(s) -y(s)|
\end{split}
\end{equation}
for $t_n \le t \le t_{n+1}$. Since this is true for all such $t$, we can take the supremum of the left 
hand side and combine it with the last term on the right hand side. We find
\begin{equation}
\sup _{t_n \le t \le t_{n+1}} |x(t) -y(t)| \le 2 |x(t_n)-y(t_n)| + 
4 \sup _{t_n \le s \le t_{n+1}}\left|\frac{v(s)}{\beta}\right| .
\end{equation}

Suppose we can prove that
\begin{equation}
\sup _{t_n \le s \le t_{n+1}}\left|\frac{v(s)}{\beta}\right| \to 0
\end{equation}
with probability one as $ \beta \to \infty$, for all $n$. Let
\[
\xi _n = \sup _{t_n \le t \le t_{n+1}} |x(t) -y(t)| .
\]
Since $x(t_0)-y(t_0) = x_0-x_0=0$, by (10.12) and (10.13), $\xi _1 \to 0$ as $\beta \to \infty$. By 
induction, it follows from (10.12) and (10.13) that $\xi _n \to 0$ for all $n$, which is what we wish 
to prove. Therefore, we need only prove (10.13).

If we regard the $x(t)$ as being known, (10.3) is an inhomogeneous linear equation for $v$, 
so that by Theorem 8.2,
\begin{equation}
\begin{split}
v(t) = e^{- \beta (t-t_n)} v(t_n) & + \beta \int ^t _{t_n} e^{- \beta (t-s)} b\big(x(s)\big)\,ds \\
& + \beta \int ^t _{t_n} e^{- \beta (t-s)} dw(s) .
\end{split}
\end{equation}
Now consider (10.8). Since
\begin{equation}
\big|b\big(x(s)\big)\big| \le \big|b\big(x(t_n)\big)\big| + \kappa |x(s)-x(t_n)|,
\end{equation}
it follows from (10.8) that
\begin{equation}
\begin{split}
|x(t) &-x(t_n)| \le \left|\frac{v(t_n)}{\beta}\right|+\left|\frac{v(t)}{\beta}\right|+(t-t_n)\big|b\big(x(t_n)\big)\big|\\
& + (t-t_n) \kappa \sup _{t_n \le s \le t_{n+1}} |x(s) -x(t_n)|+|w(t)-w(t_n)|.
\end{split}
\end{equation}
Remembering that $(t-t_n) \le 1/2 \kappa $ for $t_n \le t \le t_{n+1}$, we find as before that
\begin{equation}
\begin{split}
& \sup _{t_n \le t \le t_{n+1}}|x(t)-x(t_n)| \le 4 \sup _{t_n \le s \le t_{n+1}}\left|\frac{v(t)}{\beta}\right| \\
& \quad +\frac{1}{\kappa}\big|b\big(x(t_n)\big)\big|+ 2 \sup _{t_n \le t \le t_{n+1}} |w(t)-w(t_n)| .
\end{split}
\end{equation}
From (10.15) and (10.17), we can bound $b\big(x(s)\big)$ for $t_n \le s \le t_{n+1}$:
\begin{equation}
\begin{split}
& \sup _{t_n \le s \le t_{n+1}}\big|b\big(x(s)\big)\big| \le 2 \big|b\big(x(t_n)\big)\big| \\
& \qquad + 4 \kappa \sup _{t_n \le t \le t_{n+1}} \left|\frac{v(t)}{\beta}\right| 
+ 2 \kappa \sup _{t_n \le t \le t_{n+1}} |w(t)-w(t_n)| .
\end{split}
\end{equation}
If we apply this to (10.14) and observe that
\[
\beta \int ^t _{t_n} e^{- \beta (t-s)} ds \le 1,
\]
we obtain
\begin{equation}
\begin{split}
& \sup _{t_n \le t \le t_{n+1}} |v(t)| \le |v(t_n)|+ 2 \big|b\big(x(t_n)\big)\big| \\
& \qquad + 4 \kappa \sup _{t_n \le t \le t_{n+1}} \left|\frac{v(t)}{\beta}\right| 
+ 2 \kappa \sup _{t_n \le t \le t_{n+1}} |w(t)-w(t_n)| \\
& \qquad + \sup _{t_n \le t \le t_{n+1}} \left| \beta \int ^t _{t_n} e^{- \beta (t-s)} dw(s)\right| .
\end{split}
\end{equation}

Now choose $\beta$ so large that
\begin{equation}
\frac{4 \kappa}{\beta} \le \frac 12 ;
\end{equation}
i.e., let $\beta \ge 8 \kappa$. Then (10.19) implies that
\begin{equation}
\begin{split}
& \sup _{t_n \le t \le t_{n+1}} |v(t)| \le 2|v(t_n)| + 4 |b\big(x(t_n)\big)| \\
& \qquad +4 \kappa \sup _{t_n \le t \le t_{n+1}} |w(t)-w(t_n)| \\
& \qquad +2 \sup _{t_n \le t \le t_{n+1}} | \beta \int ^t _{t_n} e^{- \beta (t-s)} dw(s)| .
\end{split}
\end{equation}

Let
\begin{equation}
\eta _n = \sup _{t_n \le t \le t_{n+1}} \left|\frac{v(t)}{\beta}\right|,
\end{equation}
\begin{equation}
\zeta _n = \left|\frac{b\big(x(t_n)\big)}{\beta}\right|,
\end{equation}
\begin{equation}
\begin{split}
\varepsilon _n = & 2 \sup _{t_n \le t \le t_{n+1}} \left| \int ^t _{t_n} e^{- \beta (t-s)} dw(s)\right| \\
& \qquad + \frac{4 \kappa}{\beta} \sup _{t_n \le t \le t_{n+1}} |w(t)-w(t_n)| .
\end{split}
\end{equation}
Recall that our task is to show that $\eta _n \to 0$ with probability one for all $n$ as 
$\beta \to \infty$. Suppose we can show that 
\begin{equation}
\varepsilon _n \to 0
\end{equation}
with probability one for all $n$ as $\beta \to \infty$. By (10.21),
\begin{equation}
\eta _n \le 2 \eta _{n-1} + 4 \zeta _n + \varepsilon _n
\end{equation}
where $\eta _{-1}=|v_0| / \beta$, and by (10.18) for $n-1$ and (10.20),
\begin{equation}
\zeta _n \le 2 \zeta _{n-1} + \frac 12 \eta _{n-1} + \frac 12 \varepsilon _n .
\end{equation}
Now $\zeta _0 = |b(x_0)| / \beta \to 0$ and $\eta _{-1}=|v_0| / \beta \to 0$, $\zeta _1 \to 0$ by 
(10.27) and (10.25), and consequently $\eta _1 \to 0$. By induction, $\zeta _n \to 0$ and 
$\eta _n \to 0$ for all $n$. Therefore, we need only prove (10.25).

It is clear that the second term on the right hand side of (10.24) converges to 0 with probability one 
as $\beta \to \infty$, since $w$ is continuous with probability one. Let
\[
z(t) =
\begin{cases}
w(t)-w(t_n), &  t \ge t_n, \\
0, &  t < t_n .
\end{cases}
\]
Then
\begin{align*}
\int ^t _{t_n} e^{- \beta (t-s)} dw(s) & = \int ^t _{- \infty} e^{- \beta (t-s)} dz(s) \\
& =-\beta \int ^t _{\infty} e^{- \beta (t-s)} z(s)\, ds + z(t) .
\end{align*}
This converges to 0 uniformly for $t_n \le t \le t_{n+1}$ with probability one, since $z$ is 
continuous with probability one. Therefore (10.25) holds. ~~\textrm{QED}. \\

A possible physical objection to the theorem is that the initial velocity $v_0$ should not be held 
fixed as $\beta$ varies but should have a Maxwellian distribution (Gaussian with mean 0 and variance 
$D \beta$). Let $v_{00}$ have a Maxwellian distribution for a fixed value $\beta = \beta _0$. 
Then $v_0 = ( \beta / \beta _0) ^{\frac 12} v_{00}$ has a Maxwellian distribution for all $\beta$. 
Since it is still true that $v_0 / \beta \to 0$ as $\beta \to \infty$, the theorem remains true with 
a Maxwellian initial velocity.

Theorem 10.1 has a corollary that can be expressed purely in the language of partial equations: \\

P{\ninerm SEUDOTHEOREM} 10.2 
\textit{Let $b: \Bbb{R}^{\ell} \to \Bbb{R} ^{\ell}$ satisfy a global 
Lipschitz condition, and let $D$ and $\beta$ be strictly positive constants. Let $f_0$ be a 
bounded continuous function on $\Bbb{R}^{\ell}$. Let $f$ on $[0, \infty) \times \Bbb{R}^{\ell}$ 
be the bounded solution of 
\begin{equation}
\frac{\partial}{\partial t} f(t,x) = \big(D \Delta _x + b(x) \cdot \nabla _x\big) f(t,x); \qquad f(0,x) = f_0 (x) .
\end{equation}
Let $g_{\beta}$ on $[0, \infty) \times \Bbb{R}^{\ell} \times \Bbb{R}^{\ell}$ be the bounded solution of
\[
\frac{\partial}{\partial t} g_{\beta} (t,x,v) = 
\big(\beta ^2 D \Delta _v +v \cdot \nabla _x + \beta (b(x)-v) \cdot \nabla _v\big) g_{\beta} (t,x,v) ;
\]
\vskip-30pt
\begin{equation}
\hskip3truein g_{\beta} (0,x,v) = f_0 (x) .
\end{equation}
Then for all $t$, $x$, and $v$,
\begin{equation}
\lim _{\beta \to \infty} g_{\beta} (t,x,v) = f(t,x) .
\end{equation}
}

To prove this, notice that $f(t,x_0) = \mathrm{E} f_0 \big(y(t)\big)$ and $g_{\beta} (t,x_0,v_0) = 
\mathrm{E} f_0 \big(x(t)\big)$, since (10.28) and (10.29) are the backward Kolmogorov equations of the 
two processes. The result follows from Theorem 10.1 and the Lebesgue dominated convergence theorem.

There is nothing wrong with this proof---only the formulation of the result is at fault. 
Equation (10.28) is a parabolic equation with smooth coefficients, and it is a classical result that it 
has a unique bounded solution. However, (10.29) is not parabolic (it is of first order in $x$), so 
we do not know that it has a unique bounded solution. One way around this problem would be to let 
$g_{\beta, \varepsilon}$ be the unique bounded solution of (10.29) with the additional operator 
$\varepsilon \Delta _x$ on the right hand side and to prove that $g_{\beta, \varepsilon} (t,x_0,v_0) 
\to g_{\beta} (t,x_0,v_0) = \mathrm{E} f_0 \big(x(t)\big)$ as $\varepsilon \to 0$. This would give us 
a characterization of $g_{\beta}$ purely in terms of partial differential equations. We shall not do this.

%~\\[1cm]
\begin{center}
\textrm{\bf Reference}
\end{center}
[26]. Eugen Kappler, \emph{Versuche zur Messung der Avogadro-Loschmidt\-schen Zahl aus der Brownschen Bewegung einer Drehwaage}, Annalen der Physik, \textrm{11} (1931), 233--256.

%\newpage

\fancyhead[CO]{KINEMATICS OF STOCHASTIC MOTION}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Kinematics of stochastic motion}

\hspace{6mm}We shall investigate the kinematics of motion in which chance plays a r\^ole (stochastic motion).

Let $x(t)$ be the position of a particle at time $t$. What does it mean to say that the particle has 
a velocity $\dot{x} (t)$? It means that if $\Delta t$ is a very short time interval then
\[
x(t+ \Delta t)-x(t) = \dot{x} (t) \Delta t + \varepsilon,
\]
where $\varepsilon$ is a very small percentage error. This is an assumption about actual motion 
of particles that may not be true. Let us be conservative and suppose that it is not necessarily 
true. (``Conservative'' is a useful word for mathematicians. It is used when introducing a hypothesis 
that a physicist would regard as highly implausible.)

The particle should have some tendency to persist in uniform rectilinear motion for very small 
intervals of time. Let us use $\mathrm{D}x(t)$ to denote the best prediction we can make, given any relevant 
information available at time $t$, of
\[
\frac{x(t+ \Delta t)-x(t)}{\Delta t}
\]
for infinitely small positive $\Delta t$.

Let us make this notion precise.

Let $I$ be an interval that is open on the right, let $x$ be an $\Bbb{R}^{\ell}$-valued stochastic 
process indexed by $I$, and let $\mathscr{P}_t$ for $t$ in $I$ be an increasing family of $\sigma$-
algebras such that each $x(t)$ is $\mathscr{P}_t$-measurable. (This implies that $\mathscr{P}_t$ 
contains the $\sigma$-algebra generated by the $x(s)$ with $s \le t$, $s \in I$. Conversely, this 
family of $\sigma$-algebras satisfies the hypotheses.) We shall have occasion to introduce various 
regularity assumptions, denoted by (R0), (R1), etc. \\

(R0).\ Each $x(t)$ is in $\mathscr{L}^1$ and $t \mapsto x(t)$ is continuous from $I$ 
into $\mathscr{L}^1$. \\

This is a very weak assumption and by no means implies that the sample functions (trajectories) of 
the $x$ process are continuous. \\

(R1).\ The condition (R0) holds and for each $t$ in $I$,
\[
\mathrm{D}x(t) = \lim _{\Delta t \to 0+} \mathrm{E} \left\{ \frac{x(t+ \Delta t)-x(t)}{\Delta t} \,\big |\, 
\mathscr{P}_t \right\}
\]
exists as a limit in $\mathscr{L}^1$, and $t \mapsto \mathrm{D}x(t)$ is continuous 
from $I$ into $\mathscr{L}^1$. \\

Here $\mathrm{E} \{~| \mathscr{P}_t\}$ denotes the conditional expectation; cf.\ 
Doob [15, \S6]. The notation $\Delta t \to 0+$ means that $\Delta t$ tends to 0 through positive 
values. The random variable $\mathrm{D}x(t)$ is automatically $\mathscr{P}_t$-measurable. It is called the 
\emph{mean forward derivative} (or \emph{mean forward velocity} if $x(t)$ represents position).

As an example of an (R1) process, let $I=(- \infty, \infty)$, let $x(t)$ be the position in the 
Ornstein-Uhlenbeck process, and let $\mathscr{P}_t$ be the \hbox{$\sigma$-algebra} generated by the 
$x(s)$ with $s \le t$. Then $\mathrm{D}x(t) = dx(t) /dt=v(t)$. In fact, if $t \mapsto x(t)$ has a continuous strong 
derivative $dx(t)/dt$ in $\mathscr{L}^1$, then $\mathrm{D}x(t) = dx(t)/dt$. A second example of an (R1) process 
is a process $x(t)$ of the form discussed in Theorem 8.1, with $I = [0, \infty)$, $x(0)=x_0$, and 
$\mathscr{P}_t$ the \hbox{$\sigma$-algebra} generated by the $x(s)$ with $0 \le s \le t$. In this case 
$\mathrm{D}x(t) = b\big(x(t)\big)$. The derivative $dx(t)/dt$ does not exist in this example unless $w$ is identically 0. 
For a third example, let $P^t$ be a Markovian semigroup on a locally compact Hausdorff space $X$ 
with infinitesimal generator $A$, let $I=[0, \infty)$, let $\xi (t)$ be the $X$-valued random variables 
of the Markov process for some initial measure, and let $\mathscr{P}_t$ be the \hbox{$\sigma$-algebra}
generated by the $\xi (s)$ with $0 \le s \le t$. If $f$ is in the domain of the infinitesimal generator 
$A$ then $x(t) = f\big( \xi (t)\big)$ is an (R1) process, and $\mathrm{D}f\big( \xi (t)\big)= Af\big( \xi (t)\big)$.
\\
\eject

%\begin{thm}
T{\ninerm HEOREM} 11.1 {\it
Let $x$ be an {\rm (R1)} process, and let $a \le b$, $a \in I$, $b \in I$. Then
\begin{equation}
\mathrm{E} \{ x(b)-x(a) \,\big |\, \mathscr{P}_a \} = 
\mathrm{E} \left\{ \int ^b _a \mathrm{D}x(s)\,ds \,\big |\, \mathscr{P}_a \right\}
\end{equation}
}
%\end{thm}

Notice that since $s \mapsto \mathrm{D}x(s)$ is continuous in $\mathscr{L}^1$, the integral exists as a Riemann 
integral in $\mathscr{L}^1$. \\

{\it Proof.} Let $\varepsilon >0$ and let $J$ be the set of all $t$ in $[a,b]$ such that
\begin{equation}
\left\| \mathrm{E} \{ x(s) -x(a) \,|\, \mathscr{P}_a \}-
\mathrm{E} \left\{ \int ^s _a \mathrm{D}x(r)\,dr \,\big |\, \mathscr{P}_a \right\} \right\| _1 \le \varepsilon (s-a)
\end{equation}
for all $a \le s \le t$, where $\|~\| _1$ denotes the $\mathscr{L}^1$ norm. Clearly, $a$ is in $J$, 
and $J$ is a closed subinterval of $[a,b]$. Let $t$ be the right end-point of $J$, and suppose that $t <b$. 
By the definition of $\mathrm{D}x(t)$, there is a $ \delta >0$ such that $t+ \delta \le b$ and
\[
\| \mathrm{E} \{ x(t + \Delta t)-x(t) \,|\, \mathscr{P}_t \}-\mathrm{D}x(t) \Delta t \| _1 \le 
\frac{\varepsilon}{2} \Delta t
\]
for $0 \le \Delta t \le \delta$. Since conditional expectations reduce the $\mathscr{L}^1$ norm 
and since $\mathscr{P}_t \cap \mathscr{P}_a =\mathscr{P}_a$,
\begin{equation}
\| \mathrm{E} \{ x(t + \Delta t)-x(t) \,|\, \mathscr{P}_a \}-
\mathrm{E} \{ \mathrm{D}x(t) \Delta t \,|\, \mathscr{P}_a \}\| _1 \le 
\frac{\varepsilon}{2} \Delta t
\end{equation}
for $0 \le \Delta t \le \delta$. By reducing $\delta$ if necessary, we find
\[
\left\| \mathrm{D}x(t) \Delta t-\int ^{t+ \Delta t} _{t} \mathrm{D}x(s)\,ds \right\| _1 \le \frac{\varepsilon}{2} \Delta t
\]
for $0 \le \Delta t \le \delta$, since $s \mapsto \mathrm{D}x(s)$ is $\mathscr{L}^1$ continuous. Therefore,
\begin{equation}
\left\| \mathrm{E} \{ \mathrm{D}x(t) \Delta t \,\big |\, \mathscr{P}_a \}-
\mathrm{E} \left\{ \int ^{t+ \Delta t} _{t} \mathrm{D}x(s)\,ds \,\big |\, \mathscr{P}_a \right\}\right\| _1 \le 
\frac{\varepsilon}{2} \Delta t
\end{equation}
for $0 \le \Delta t \le \delta$. From (11.2) for $s=t$, (11.3), and (11.4), it follows that (11.2) folds for 
all $t + \Delta t$ with $0 \le \Delta t \le \delta$. This contradicts the assumption that $t$ is the 
end-point of $J$, so we must have $t=b$. Since $\varepsilon$ is 
arbitrary, (11.1) holds. ~~\textrm{QED}. \\

Theorem 11.1 and its proof remain valid without the assumption that $x(t)$ is 
$\mathscr{P}_t$-measurable.
\\

%\begin{thm}
T{\ninerm HEOREM} 11.2 {\it
An {\rm (R1)} process is a martingale if and only if $\mathrm{D}x(t)=0$, $t \in I$. It is a submartingale 
if and only if $\mathrm{D}x(t) \ge 0$, $t \in I$ and a supermartingale if and only if $\mathrm{D}x(t) \le 0$, $t \in I$.
}
\\
%\end{thm}

We mean, of course, martingale, etc., relative to the $\mathscr{P}_t$. This theorem is an immediate 
consequence of Theorem 11.1 and the definitions (see Doob [15, p.\ 294]). Note that in the older 
terminology, ``semimartingale'' means submartingale and ``lower semimartingale'' means 
supermartingale.

Given an (R1) process $x$, define the random variable $y(a,b)$, for all $a$ and $b$ in $I$, by
\begin{equation}
x(b)-x(a) = \int ^b _a \mathrm{D}x(s)\,ds + y(a,b) .
\end{equation}
We always have $y(b,a)=-y(a,b)$, $y(a,b)+y(b,c)=y(a,c)$, and $y(a,b)$ is $\mathscr{P}_{\max (a,b)}$-
measurable, for all $a$, $b$, and $c$ in $I$. We call a stochastic process indexed by $I \times I$ 
that has these three properties a \emph{difference process}. If $y$ is a difference process, we 
can choose a point $a_0$ in $I$, define $y(a_0)$ arbitrarily (say $y(a_0)=0$), and define $y(b)$ for 
all $b$ in $I$ by $y(b)=y(a_0,b)$. Then $y(a,b)=y(b)-y(a)$ for all $a$ and $b$ in $I$. The only trouble 
is that $y(b)$ will not in general be $\mathscr{P}_b$-measurable for $b < a_0$. If $I$ has a left 
end-point, we can choose $a_0$ to be it and then $y(b)$ will always be \hbox{$\mathscr{P}_b$-measurable}. 
By Theorem 11.1, $\mathrm{E} \{ y(b)-y(a)  | \mathscr{P}_a \} = 0$ whenever $a \le b$, so that 
when $a_0$ is the left end-point of $I$, $y(b)$ is a martingale relative to the $\mathscr{P}_b$. 
In the general case, we call a difference process $y(a,b)$ such that 
$\mathrm{E} \{ y(a,b)  \,|\, \mathscr{P}_a \} = 0$ whenever $a \le b$ a \emph{difference martingale}. 
The following is an immediate consequence of Theorem 11.1.
\\

%\begin{thm}
T{\ninerm HEOREM} 11.3 {\it
Let $x$ be an {\rm (R1)} process, and define $y$ by {\rm (11.5)}. Then $y$ is a difference 
martingale.
}
\\
%\end{thm}

From now on we shall write $y(b)-y(a)$ instead of $y(a,b)$ when $y$ is a difference process.

We introduce another regularity condition, denoted by (R2). It is a regularity condition on a difference 
martingale $y$. If it holds, we say that $y$ is an (R2) difference martingale, and if in addition $y$ is 
defined in terms of an (R1) process $x$ by (11.5) then we say that $x$ is an (R2) process. \\

(R2).\ For each $a$ and $b$ in $I$, $y(b)-y(a)$ is in $\mathscr{L}^2$. For each $t$ in $I$,
\begin{equation}
\sigma ^2 (t)= \lim _{ \Delta t \to 0+} \mathrm{E} \left\{ \frac{[y(t+ \Delta t)-y(t)]^2}{ \Delta t} 
\,\big |\, \mathscr{P}_t \right\}
\end{equation}
exists in $\mathscr{L}^1$, and $t \mapsto \sigma ^2(t)$ is continuous from $I$ into $\mathscr{L}^1$. \\

The process $y$ has values in $\Bbb{R}^{\ell}$. In case $\ell >1$, we understand the expression 
$[y(t+ \Delta t)-y(t)]^2$ to mean $[y(t+ \Delta t)-y(t)] \otimes [y(t+ \Delta t)-y(t)]$, and 
$\sigma ^2(t)$ is a matrix of positive type.

Observe that $\Delta t$ occurs to the first power in (11.6) while the term $[y(t+ \Delta t)-y(t)]$ occurs to 
the second power.
\\

%\begin{thm}
T{\ninerm HEOREM} 11.4 {\it
Let $y$ be an {\rm (R2)} difference martingale, and let $a \le b$, $a \in I$, $b \in I$. Then
\begin{equation}
\mathrm{E} \{ [y(b)-y(a)]^2 \, |\, \mathscr{P}_a \} = 
\mathrm{E} \left\{ \int ^b _a \sigma ^2 (s)\,ds \,\big |\, \mathscr{P}_a \right\} .
\end{equation}
}
%\end{thm}

The proof is so similar to the proof of Theorem 11.1 that it will be omitted.

Next we shall discuss the It\^o-Doob stochastic integral, which is a generalization of the 
Wiener integral. The new feature is that the integrand is a random variable depending on the 
past history of the process.

Let $y$ be an (R2) difference martingale. Let $\mathscr{H}_0$ be the set of functions of the form
\begin{equation}
f = \sum _{i=1} ^n f_i \chi _{[a_i,b_i ]},
\end{equation}
where the interval $[a_i,b_i]$ are non-overlapping intervals in $I$ and each $f_i$ is a real-valued 
$\mathscr{P}_{a_i}$-measurable random variable in $\mathscr{L}^2$. (The symbol $\chi$ denotes the 
characteristic function.) Thus each $f$ in $\mathscr{H}_0$ is a stochastic process indexed by $I$. 
For each $f$ given by (11.8) we define the stochastic integral
\[
\int f(t)\, dy(t) = \sum _{i=1}^{n} f_i [y(b_i)-y(a_i)] .
\]
This is a random variable.

For $f$ in $\mathscr{H}_0$,
\begin{equation}
\mathrm{E} \left[ \int f(t)\, dy(t)\right]^2 = \sum _{i,j=1} ^n \mathrm{E} \,f_i\, [y(b_i) -y(a_i)]\, f_j\, [y(b_j)-y(a_j)] .
\end{equation}
If $i<j$ then $f_i\, [y(b_i) -y(a_i)]\, f_j$ is $\mathscr{P}_{a_j}$-measurable, and 
$$\mathrm{E} \{ y(b_j)-y(a_j) \, |\, \mathscr{P}_{a_j} \} = 0$$
since $y$ is a difference martingale. 
Therefore the terms with $i<j$ in (11.9) are 0, and similarly for the terms with $i>j$. The terms 
with $i=j$ are
\[
\mathrm{E} f_i ^2 \,\mathrm{E} \left\{ \int ^{b_i} _{a_i} \sigma ^2(s)\, ds \,\big |\, \mathscr{P}_{a_i} \right\} =
\int ^{b_i} _{a_i} \mathrm{E} f_i ^2 \sigma ^2(s)\, ds
\]
by (11.7). Therefore
\[
\mathrm{E} \left[ \int f(t)\, dy(t)\right]^2 = \int _{I} \mathrm{E} f^2 (t) \sigma ^2(t)\, dt .
\]
This is a matrix of positive type. If we give $\mathscr{H}_0$ the norm
\begin{equation}
\| f \| ^2 = \operatorname{tr} \int _{I} \mathrm{E} f^2 (t) \sigma ^2(t)\, dt
\end{equation}
then $\mathscr{H}_0$ is a pre-Hilbert space, and the mapping $f \mapsto \int f(t)\, dy(t)$ is isometric from 
$\mathscr{H}_0$ into the real Hilbert space of square-integrable \hbox{$\Bbb{R}^{\ell}$-valued} 
random variables, which will be denoted by $\mathscr{L}^{2}\big(I;{\Bbb{R}^{\ell}}\big)$.

Let $\mathscr{H}$ be the completion of $\mathscr{H}_0$. The mapping $f \mapsto \int f(t)\, dy(t)$ 
extends uniquely to be unitary from $\mathscr{H}$ into $\mathscr{L}^{2}\big(I;{\Bbb{R}^{\ell}}\big)$. 
Our problem now is to describe $\mathscr{H}$ in concrete terms.

Let $\sigma (t)$ be the positive square root of $\sigma ^2(t)$. If $f$ is in $\mathscr{H}_0$ 
then $f \sigma$ is square-integrable. If $f_j$ is a Cauchy sequence in $\mathscr{H}_0$ then 
$f_j \sigma$ converges in the $\mathscr{L}^2$ norm, so that a subsequence, again denoted by $f_j$, 
converges a.e. to a square-integrable matrix-valued function $g$ on $I$. Therefore $f_j$ converges 
for a.e. $t$ such that $\sigma (t) \ne 0$. Let us define $f(t) = \lim f_j(t)$ when the limit exists and 
define $f$ arbitrarily to be 0 when the limit does not exist. Then $\| f_j-f \| \to 0$, and 
$f(t) \sigma (t)$ is a $\mathscr{P}_t$-measurable square-integrable random variable for a.e. $t$. 
By definition of strong measurability [14, \S5], $f \sigma$ is strongly measurable. Let $\mathscr{K}$ 
be the set of all functions $f$, defined a.e. on~$I$, such that $f \sigma$ is a strongly measurable 
square-integrable function with $f(t)$ $\mathscr{P}_t$-measurable for a.e. $t$. We have seen that 
every element of $\mathscr{H}$ can be identified with an element of $\mathscr{K}$, uniquely defined 
except on sets of measure 0.

Conversely, let $f$ be in $\mathscr{K}$. We wish to show that it can be approximated arbitrarily 
closely in the norm (11.10) by an element of $\mathscr{H}_0$. Firstly, $f$ can by approximated 
arbitrarily closely by an element of $\mathscr{K}$ with support contained in a compact interval $I_0$ 
in $I$, so we may as well assume that $f$ has support in $I_0$. Let
\[
f_k(t)=
\begin{cases}
\quad k, &  \quad \; f(t) \; > \quad k, \\
f(t), &  \quad |f(t)| \le \quad k, \\
-k, &  \quad \; f(t) \; < \, -k.
\end{cases}
\]
Then $\| f_k -f \| \to 0$ as $k \to \infty$, so we may as well assume that $f$ is uniformly bounded 
(and consequently has uniformly bounded $\mathscr{L}^{2}$ norm). Divide $I_0$ into $n$ equal parts, 
and let $f_n$ be the function that on each subinterval is the average (Bochner integral [14, \S5]) of 
$f$ on the preceding subinterval (and let $f_n$ be 0 on the first subinterval). Then $f_n$ is in 
$\mathscr{H}_0$ and $\| f_k -f \| \to 0$.

With the usual identification of functions equal a.e., we can identify $\mathscr{H}$ and $\mathscr{K}$. 
We have proved the following theorem.
\\

%\begin{thm}
T{\ninerm HEOREM} 11.5 {\it
Let $\mathscr{H}$ be the Hilbert space of functions $f$ defined a.e. on $I$ such that $f \sigma$ is 
strongly measurable and square-integrable and such that $f(t)$ is $\mathscr{P}_t$-measurable for 
a.e. $t$, with the norm {\rm (11.10)}. There is a unique unitary mapping $f \mapsto \int f(y)\, dy(t)$ from 
$\mathscr{H}$ into $\mathscr{L}^{2} \big(I;{\Bbb{R}^{\ell}}\big)$ 
such that if $f = f_0 \chi _{[a,b]}$ where 
$a \le b, a \in I, b \in I, f_0 \in \mathscr{L}^{2}, f_0$ $ \mathscr{P}_a$-measurable, then
\[
\int f(y)\, dy(t) = f_0 [y(b)-y(a)] .
\]
}
%\end{thm}

We now introduce our last regularity hypothesis. \\

(R3).\ For a.e. $t$ in $I$, $\det \sigma ^2(t) >0$ a.e. \\

An (R2) difference martingale for which this holds will be called an (R3) difference martingale. 
An (R2) process $x$ for which the associated difference martingale $y$ satisfies this will be called 
an (R3) process.

Let $\sigma ^{-1} (t) = \sigma (t)^{-1}$, where $\sigma (t)$ is the positive square root of $\sigma ^2(t)$.
\\

%\begin{thm}
T{\ninerm HEOREM} 11.6 {\it
Let $x$ be an {\rm (R3)} process. Then there is a difference martingale $w$ such that
\[
\mathrm{E} \{ [w(b)-w(a)]^2 \,\big |\, \mathscr{P}_{a} \}= b-a
\]
and
\[
x(b)-x(a) = \int ^b _a \mathrm{D}x(s)\,ds + \int ^b _a \sigma (s)\, dw(s)
\]
whenever $a \le b$, $a \in I$, $b \in I$.
}
\\
%\end{thm}

{\it Proof.} Let
\[
w(a,b) = \int ^b _a \sigma ^{-1} (s)\, dy(s) .
\]
This is well defined, since each component of $\sigma ^{-1} \chi _{[a,b]}$ is in $\mathscr{H}$. 
If $f$ is in $\mathscr{H}_0$, a simple computation shows that $\int ^b _a f(s)\,dy(s)$ is a difference martingale, so the same is true if $f$ is in $\mathscr{H}$ or if each $f \chi _{[a,b]}$ is in $\mathscr{H}$. 
Therefore, $w$ is a difference martingale, and we will write $w(b)-w(a)$ for $w(a,b)$.

If $f$ is in $\mathscr{H}_0$, given by (11.8), and if $f(t)=0$ for all $t<a$, then
\begin{align*}
& \mathrm{E} \left\{ \left[ \int f(t)\, dy(t)\right]^2 \,\big |\, \mathscr{P}_{a} \right\} = \\
& \mathrm{E} \left\{ \sum _i f_i ^2 \,[y(b_i) -y(a_i)]^2 \,\big |\, \mathscr{P}_{a} \right\} =\\
& \mathrm{E} \left\{ \mathrm{E} \left\{ \sum _i f_i ^2 \int ^{b_i} _{a_i} \sigma ^2(s)\,ds 
\,\big |\, \mathscr{P}_{a_i} \right\} \,\big |\, \mathscr{P}_{a} \right\}= \\
& \mathrm{E} \left\{ \sum _i f_i ^2 \int ^{b_i} _{a_i} \sigma ^2(s)\,ds \,\big |\, \mathscr{P}_{a} \right\}= \\
& \mathrm{E} \left\{ \int f^2(s) \sigma ^2(s)\, ds \,\big |\, \mathscr{P}_{a} \right\} .
\end{align*}
By continuity,
\begin{align*}
& \mathrm{E} \left\{ \left[ \int ^b _a f(t)\, dy(t)\right]^2 \,\big |\, \mathscr{P}_{a} \right\} = \\
& \mathrm{E} \left\{ \int ^b _a f^2(s) \sigma ^2(s)\, ds \,\big |\, \mathscr{P}_{a} \right\}
\end{align*}
for all $f$ in $\mathscr{H}$. If we apply this to the components of $\sigma ^{-1} \chi _{[a,b]}$ 
we find
\begin{align*}
& \mathrm{E} \{ [ w(b)-w(a)]^2 \, |\, \mathscr{P}_{a} \} = \\
& \mathrm{E} \left\{ \int ^b _a \sigma ^{-1} (s) \sigma ^2(s) \sigma ^{-1} (s)\,ds \,\big |\, \mathscr{P}_{a} \right\} = \\
& b-a,
\end{align*}
whenever $a \le b$, $a \in I$, $b \in I$. Consequently, $w$ is an (R2) 
$\big($in fact, (R3)$\big)$ difference 
martingale, and the corresponding $\sigma ^2$ is identically 1. Therefore we can construct 
stochastic integrals with respect to $w$.

Formally, $\,dw(t) = \sigma ^{-1} (t)\, dy(t)$, so that $dy(t) = \sigma (t) \,dw(t)$. Let us prove that, in fact, 
\[
y(b)-y(a) = \int ^b _a \sigma (s)\, dw(s) .
\]
A simple calculation shows that if $f$ is in $\mathscr{H}_0$ then
\[
\int f(s)\, dw(s) = \int f(s) \sigma ^{-1} (s)\, dy(s) .
\]
Consequently, the same holds for any $f$ in $\mathscr{H}$. Therefore,
\begin{align*}
\int ^b _a \sigma (s)\, dw(s) & = \int ^b _a \sigma (s) \sigma ^{-1} (s)\, dy(s) \\
& = y(b)-y(a) = y(a,b) .
\end{align*}
The theorem follows from the definition (11.5) of $y$. ~~\textrm{QED}. \\

It is possible that the theorem remains true without the regularity assumption (R3) provided that one 
is allowed to enlarge the underlying probability space and the $\sigma$-algebras $\mathscr{P}_t$.

A fundamental aspect of motion has been neglected in the discussion so far; to wit, the continuity of 
motion. We shall assume from now on that (with probability one) the sample functions of $x$ are 
continuous. By~(11.5), this means that the same functions of $y$ are continuous. (Use $\omega$ to denote 
a point in the underlying probability space. We can choose a version $\mathrm{D}x( s, \omega)$ of the 
stochastic process $\mathrm{D}x$ that is jointly measurable in $s$ and $\omega$, 
since $s \mapsto \mathrm{D}x(s)$ is 
continuous in $\mathscr{L}^1$ [15, \S6, p.\ 60 ff]. 
Then $\int ^b _a \mathrm{D}x(s, \omega)\, ds$ is in fact 
absolutely continuous as $b$ varies, so that $y(b)-y(a)$ has continuous sample paths as $b$ varies.) 
Next we show (following Doob [15, p.\ 446]) that this implies that $\omega$ has continuous sample 
functions.
\\

%\begin{thm}
T{\ninerm HEOREM} 11.7 {\it
Let $y$ be an {\rm (R2)} difference martingale whose sample functions are continuous with 
probability one, and let $f$ be in $\mathscr{H}$. let
\[
z(b)-z(a) = \int ^b _a f(s)\, dy(s) .
\]
Then $z$ is a difference martingale whose sample paths are continuous with probability one.
}
\\
%\end{thm}

{\it Proof.} If $f$ in in $\mathscr{H}_0$, this is evident. If $f$ is in $\mathscr{H}$, 
let $f_n$ be in $\mathscr{H}_0$ with $\| f-f_n \| \le 1/n^2 $, where the norm is given by (11.10). Let
\[
z_n(b)-z_n(a) = \int ^b _a f_n(s)\, dy(s) .
\]
Then $z-z_n$ is a difference martingale. (We already observed in the proof of Theorem 11.6 that $z$ 
is a difference martingale---only the continuity of sample functions is at issue.)

By the Kolmogorov inequality for martingales (Doob [15, p.\ 105]), if $S$ is any finite subset of 
$[a,b]$,
\[
\mathrm{Pr} \left\{\sup _{s \in S} |z(s) -z_n(s) |> \frac 1n \right\} \le \frac{1}{n^4} \cdot n^2 = \frac{1}{n^2}.
\]
Since $S$ is arbitrary, we have
\[
\mathrm{Pr} \left\{\sup _{a \le s \le b} |z(s) -z_n(s) |> \frac 1n \right\} \le \frac{1}{n^2} .
\]
(This requires a word concerning interpretation, since the supremum is over an uncountable set. 
We can either assume that $z-z_n$ is separable in the sense of Doob or take the product space 
representation as in [25, \S9] of the pair $z$, $z_n$.) By the Borel-Cantelli lemma, $z$ converges 
uniformly on $[a,b]$ to $z$. ~~\textrm{QED}. \\

Notice that we only need $f$ to be locally in $\mathscr{H}$; i.e., we only need $f \chi _{[a,b]}$ to 
be in $\mathscr{H}$ for $[a,b]$ any compact subinterval of $I$. In particular, if $y$ is an (R3) process 
the result above applies to each component of $\sigma ^{-1}$, so that $w$ has continuous sample 
paths if $y$ does.

Now we shall study the difference martingale $w$ (with $ \sigma ^2$ identically~1) under the 
assumption that $w$ has continuous sample paths.
\\

%\begin{thm}
T{\ninerm HEOREM} 11.8 {\it
Let $w$ be a difference martingale in $\Bbb{R}^{\ell}$ satisfying
\[
\mathrm{E} \{ [ w(b)-w(a)]^2 \, |\, \mathscr{P}_{a} \} =b-a
\]
whenever $a \le b, a \in I, b \in I$, and having continuous sample paths with probability one. 
Then $w$ is a Wiener process.
}
\\
%\end{thm}

{\it Proof.} We need only show that the $w(b) -w(a)$ are Gaussian. There is no loss of generality 
in assuming that $a=0$ and $b=1$. First we assume that $\ell =1$.

Let $\Delta t$ be the reciprocal of a strictly positive integer and let $\Delta w(t) = w(t + \Delta t) 
- w(t)$. Then
\[
[w(1)-w(0) ]^n = \sum \Delta w(t_1) \ldots \Delta w(t_n),
\]
where the sum is over all $t_1, \ldots, t_n$ ranging over $0, \Delta t, 2 \Delta t, \ldots, 1- 
\Delta t$.

We write the sum as $\sum = \sum {'}+ \sum {''}$, where $\sum {'}$ 
is the sum of all terms in which no three of the $t_i$ are equal.

Let $B(K)$ be the set such that $|w(1)-w(0)| \le K$. Then
\[
\lim _{K \to \infty} \mathrm{Pr} \big(B(K)\big) = 1 .
\]
Let $\Gamma ( \varepsilon, \delta)$ be the set such that $|w(t) -w(s)| \le \varepsilon$ whenever 
$|t-s| \le \delta$, for $0 \le t$, $s \le 1$. Since $w$ has continuous sample paths with probability one,
\[
\lim _{\delta \to \infty} \mathrm{Pr} (\Gamma \big( \varepsilon, \delta)\big) = 1
\]
for each $\varepsilon >0$.

Let $\alpha >0$. Choose $K \ge 1$ so that $\mathrm{Pr} \big(B(K)\big) \ge 1- \alpha$. Given~$n$, 
choose~$\varepsilon$ so small that $n K^n \varepsilon \le \alpha$ and then choose $\delta$ so small 
that $\mathrm{Pr} (\Gamma \big( \varepsilon, \delta)\big) \ge 1-\alpha$. Now the sum $\sum {''}$ 
can be written
\[
\hbox{$\sum_0'' +\sum_1'' + \cdots +\sum_{n-3}'',$} 
\]
where $\sum_\nu''$ means that exactly $ \nu$ of the $t_i$ are distinct and some three 
of the $t_i$ are equal. Then $\sum_\nu''$ has a factor $[w(1)-w(0)]^{ \nu}$ times a 
sum of terms in which all $t_i$ that occur, occur at least twice, and in which at least one $t_i$ 
occurs at least thrice. Therefore, if $\Delta t \le \delta$,
\[
\left| \int _{\Gamma ( \varepsilon, \delta) \cap B(K)} \hbox{$\sum_\nu''$}\, d\, \mathrm{Pr} \right| \le 
K^{ \nu} \varepsilon \int \sum \Delta w(t_1)^2 \ldots \Delta w(t_j)^2 d\, \mathrm{Pr}
\le K^{ \nu} \varepsilon,
\]
where the $t_1, \ldots, t_j$ are distinct. Therefore
\[
\left| \int _{\Gamma ( \varepsilon, \delta) \cap B(K)} \hbox{$\sum''$}\, d\, \mathrm{Pr} \right| \le 
n K^n \varepsilon \le \alpha .
\]

Those terms in $\sum'$ in which one or more of the $t_i$ occurs only once have 
expectation 0, so
\[\int \hbox{$\sum'$}\, d\, \mathrm{Pr} = \mu _n,
\]
where $\mu _n =0$ if $n$ is odd and $\mu _n = (n-1)(n-3) \ldots 5 \cdot 3 \cdot 1$ if $n$ is even, 
since this is the number of ways of dividing $n$ objects into distinct pairs.

Consequently, the integral of $[w(1)-w(0)]^n$ over a set of arbitrarily large measure is arbitrarily 
close to $\mu _n$ . If $n$ is even, the integrand $[w(1)-w(0)]^n$ is positive, so this shows that 
$[w(1)-w(0)]^n$ is integrable for all even $n$ and hence for all $n$. Therefore,
\[
\mathrm{E} [w(1)-w(0)]^n = \mu _n
\]
for all $n$. But the $\mu _n$ are the moments of the Gaussian measure with mean 0 and variance 1, 
and they increase slowly enough for uniqueness to hold in the moment problem. In fact,
\begin{align*}
\mathrm{E} e^{i \lambda }[w(1)-w(0)] & = \mathrm{E} \sum ^{\infty}_{n=0} 
\frac{(i \lambda)^n}{n!} [w(1)-w(0)]^n \\
& = \sum ^{\infty}_{n=0} \frac{(i \lambda)^n}{n!} \mu _n = e^{- \frac{ \lambda ^2}{2}},
\end{align*}
so that $w(1)-w(0)$ is Gaussian.

The proof for $\ell >1$ goes the same way, except that all products are tensor products. For example, 
$(n-1)(n-3) \ldots 3 \cdot 1$ is replaced by
\[
(n-1) \delta _{i_1 i_2} (n-3) \delta _{i_3 i_4} \ldots 3 \delta _{i_{n-3} i_{n-2}} 
1 \delta _{i_{n-1} i_{n}} .
\]
~~\textrm{QED}. \\

We summarize the results obtained so far in the following theorem.
\\

%\begin{thm}
T{\ninerm HEOREM} 11.9 {\it
Let $I$ be an interval open on the right, $\mathscr{P}_t$ (for \hbox{$t \in I$}) an increasing family of 
$\sigma$-algebras of measurable sets on a probability space, $x$ a stochastic process on 
$\Bbb{R}^{\ell}$ having continuous sample paths with probability one such that each $x(t)$ is 
$\mathscr{P}_t$-measurable and such that
\[
\mathrm{D}x(t) = \lim _{\Delta t \to 0 +} \mathrm{E} \left\{ \frac{x( \Delta t + t) -x(t)}{ \Delta t} 
\,\big |\, \mathscr{P}_{t} \right\}
\]
and
\[
\sigma ^2 (t) = \lim _{\Delta t \to 0 +} \mathrm{E} \left\{ \frac{[x( \Delta t + t) -x(t)]^2}{ \Delta t} 
\,\big |\, \mathscr{P}_{t} \right\}
\]
exist in $\mathscr{L}^1$ and are $\mathscr{L}^1$ continuous in $t$, and such that $\sigma ^2 (t)$ 
is a.e. invertible for a.e. $t$. Then there is a Wiener process $w$ on $\Bbb{R}^{\ell}$ such that 
each $w(t)-w(s)$ is $\mathscr{P}_{ \max (t,s)}$-measurable, and
\[
x(b) -x(a) = \int ^b _a \mathrm{D}x(s)\,ds + \int ^b _a \sigma (s)\,dw(s)
\]
for all $a$ and $b$ in $I$.
}
\\
%\end{thm}

\begin{center}
$* \qquad * \qquad* \qquad * \qquad *$
\end{center}

So far we have been adopting the standard viewpoint of the theory of stochastic processes, that 
the past is known and that the future develops from the past according to certain probabilistic laws. 
Nature, however, operates on a different scheme in which the past and the future are on an equal 
footing. Consequently it is important to give a treatment of stochastic motion in which a complete 
symmetry between past and future is maintained.

Let $I$ be an open interval, let $x$ be an $\Bbb{R}^{\ell}$-valued stochastic process indexed by $I$, let 
$\mathscr{P}_t$ for $t$ in $I$ be an increasing family of $\sigma$-algebras such that each $x(t)$ 
is $\mathscr{P}_t$-measurable, and let $\mathscr{F}_t$ be a decreasing family of
\hbox{$\sigma$-algebras} such that each $x(t)$ is $\mathscr{F}_t$-measurable. ($\mathscr{P}_t$
represents the past, $\mathscr{F}_t$ the future.) The following regularity 
conditions make the conditions (R1), (R2), and (R3) symmetric with respect to past and future. 
The condition (R0) is already symmetric. \\

(S1).\ The condition (R1) holds and, for each $t$ in $I$,
\[
\mathrm{D}_* x(t) = \lim _{\Delta t \to 0 +} \mathrm{E} \left\{ \frac{x(t)-x(t- \Delta t)}{ \Delta t} 
\,\big |\, \mathscr{F}_{t} \right\}
\]
exists as a limit in $\mathscr{L}^1$, and $t \mapsto \mathrm{D}_* x(t)$ is continuous from $I$ 
into $\mathscr{L}^1$. \\

Notice that the notation is chosen so that if $t \mapsto x(t)$ is strongly differentiable in $\mathscr{L}^1$ 
then $\mathrm{D}x(t) = \mathrm{D}_* x(t) = dx(t)/dt$. The random variable $\mathrm{D}_* x(t)$ is called the \emph{mean backward 
derivative} or \emph{mean backward velocity}, and is in general different from $\mathrm{D}x(t)$.

We define $y_* (a,b) = y_* (b)-y_* (a)$ by
\[
x(b) -x(a) = \int ^b _a \mathrm{D}_* x(s)\,ds + y_* (b)-y_* (a) .
\]
It is a difference martingale relative to the $\mathscr{F}_t$ with the direction of time reversed. \\

(S2).\ The conditions (R2) and (S1) hold and, for each $t$ in $I$,
\[
\sigma ^2 _* (t) = \lim _{\Delta t \to 0 +} \mathrm{E} \left\{ \frac{[y(t)-y(t- \Delta t)]^2}{ \Delta t} 
\,\big |\, \mathscr{F}_{t} \right\}
\]
exists as a limit in $\mathscr{L}^1$ and $t \mapsto \sigma ^2 _* (t)$ is continuous from $I$ into 
$\mathscr{L}^1$. \\

(S3).\ The conditions (R3) and (S2) hold and $\det \sigma ^2 _* (t)>0 $ a.e. for a.e.~$t$. \\

We obtain theorems analogous to the preceding ones. In particular, if $a \le b$, $a \in I$, $b \in I$, 
then for an (S1) process
\begin{equation}
\mathrm{E} \{ x(b)-x(a) \,|\, \mathscr{F}_b \} = 
\mathrm{E} \left\{ \int ^b _a \mathrm{D}_*x(s)\,ds \,\big |\, \mathscr{F}_b \right\},
\end{equation}
and for an (S2) process
\begin{equation}
\mathrm{E} \{ [y_*(b)-y_*(a)]^2 \,|\, \mathscr{F}_b \} = 
\mathrm{E} \left\{ \int ^b _a \sigma ^2_* (s)\,ds \,\big |\, \mathscr{F}_b \right\} .
\end{equation}

%\begin{thm}
T{\ninerm HEOREM} 11.10 {\it
Let $x$ be an {\rm(S1)} process. Then
\begin{equation}
\mathrm{E} \mathrm{D}x(t) = \mathrm{E} \mathrm{D}_* x(t)
\end{equation}
for all $t$ in $I$. Let $x$ be an {\rm (S2)} process. Then
\begin{equation}
\mathrm{E} \sigma ^2(t) = \mathrm{E} \sigma ^2_*(t)
\end{equation}
for all $t$ in $I$.
}
\\
%\end{thm}

{\it Proof.} By Theorem 11.1 and (11.11), if we take absolute expectations we find
\[
\mathrm{E}[ x(b)-x(a)] = \mathrm{E} \int ^b _a \mathrm{D}x(s)\,ds=\mathrm{E} \int ^b _a \mathrm{D}_*x(s)\,ds
\]
for all $a$ and $b$ in $I$. Since $s \mapsto \mathrm{D}x(s)$ and $s \mapsto \mathrm{D}_* x(s)$ are continuous in $\mathscr{L}^1$, 
(11.13) holds. Similarly, (11.14) follows from Theorem 11.4 and (11.12). ~~\textrm{QED}.
\\

%\begin{thm}
T{\ninerm HEOREM} 11.11 {\it
Let $x$ be an {\rm (S1)} process. Then $x$ is a constant (i.e., $x(t)$ is the same random variable 
for all $t$) if and only if $\mathrm{D}x=\mathrm{D}_*x=0$.
}
\\
%\end{thm}

{\it Proof.} The only if part of the theorem is trivial. Suppose that $\mathrm{D}x=\mathrm{D}_*x=0$. By Theorem 11.2, 
$x$ is a martingale and a martingale with the direction of time reversed. Let $t_1 \ne t_2$, 
$x_1=x(t_1)$, $x_2=x(t_2)$. Then $x_1$ and $x_2$ are in $\mathscr{L}^1$ and 
$\mathrm{E} \{ x_1 | x_2 \} =x_2$, $\mathrm{E} \{ x_2 | x_1 \} =x_1$. We wish to show that $x_1 =x_2$ 
(a.e., of course).

If $x_1$ and $x_2$ are in $\mathscr{L}^2$ (as they are if $x$ is an (S2) process) there is a trivial proof, 
as follows. We have
\[
\mathrm{E} \{ (x_2-x_1)^2 | x_1 \} = \mathrm{E} \{ x_2^2-2 x_2x_1+x_1^2 | x_1 \} =
\mathrm{E} \{ x_2^2 | x_1 \} -x_1^2,
\]
so that if we take absolute expectations we find
\[
\mathrm{E} (x_2-x_1)^2 =\mathrm{E} x_2^2-\mathrm{E} x_1^2 .
\]
The same result holds with $x_1$ and $x_2$ interchanged. Thus $\mathrm{E} (x_2-x_1)^2=0$, 
$x_2=x_1$ a.e.

G. A. Hunt showed me the following proof for the general case ($x_1$, $x_2$ in $\mathscr{L}^1$).

Let $\mu$ be the distribution of $x_1$, $x_2$ in the plane. We can take $x_1$ and $x_2$ to be the 
coordinate functions. Then there is a conditional probability distribution $p(x_1, \cdot)$ such that 
if $\nu$ is the distribution of $x_1$ and $f$ is a positive Baire function on $\Bbb{R}^2$,
\[
\int f(x_1,x_2)\, d \mu (x_1,x_2) = \iint f(x_1,x_2) \,p(x_1,x_2)\,d \nu (x_1) .
\]
(See Doob [15, \S6, pp.\ 26--34].) Then
\[
\mathrm{E} \{ \varphi (x_2) \,|\, x_1 \} = \int \varphi (x_2) \,p(x_1,dx_2) \quad {\rm a.e.}\;[ \nu ]
\]
provided $\varphi (x_2)$ is in $\mathscr{L}^1$. Take $\varphi$ to be strictly convex with 
$| \varphi ( \xi) | \le | \xi |$ for all real $\xi$ (so that $\varphi (x_2)$ is in $\mathscr{L}^1$). Then, 
for each $x_1$, since $\varphi$ is strictly convex, Jensen's inequality gives
\[
\varphi \left( \int x_2 \,p(x_1,dx_2)\right) < \int \varphi (x_2) \,p(x_1,dx_2)
\]
unless $\varphi (x_1)= \int \varphi (x_2) \,p(x_1,dx_2)$ a.e. $[p(x_1, \cdot)]$. But
\[
\int x_2 \,p(x_1,dx_2) = x_1 \quad {\rm a.e.}\;[ \nu ],
\]
so, unless $x_2 = x_1$ a.e. $[ \nu ]$,
\[
\varphi (x_1) < \int \varphi (x_2) \,p(x_1,dx_2) .
\]
If we take absolute expectations, we find $\mathrm{E} \varphi (x_1) < \mathrm{E} \varphi (x_2)$ 
unless $x_2 = x_1$ a.e. The same argument gives the reverse inequality, so $x_2 = x_1$ a.e. ~~\textrm{QED}.
\\

%\begin{thm}
T{\ninerm HEOREM} 11.12 {\it
Let $x$ be and $y$ be {\rm (S1)} processes with respect to the same families of $\sigma$-algebras 
$\mathscr{P}_t$ and $\mathscr{F}_t$, and suppose that $x(t), y(t), \mathrm{D}x(t)$, $\mathrm{D}y(t)$, $\mathrm{D}_*x(t)$, and 
$\mathrm{D}_*y(t)$ all lie in $\mathscr{L}^2$ and are continuous functions of $t$ in $\mathscr{L}^2$. Then
\[
\frac{d}{dt} \mathrm{E} x(t)y(t) = \mathrm{E}\mathrm{D}x(t) \cdot y(t) + \mathrm{E} x(t) \mathrm{D}_* y(t) .
\]
}
%\end{thm}

{\it Proof.} We need to show for $a$ and $b$ in $I$, that
\[
\mathrm{E} \,[x(b)y(b)-x(a)y(a)] = \int ^b _a \mathrm{E}\,[\mathrm{D}x(t) \cdot y(t) + x(t) \mathrm{D}_* y(t) ]dt.
\]
(Notice that the integrand is continuous.) Divide $[a,b]$ into $n$ equal parts: $t_j=a+j(b-a)/n$ 
for $j=0, \ldots,n$. Then
\[
\begin{split}
& \mathrm{E} \,[x(b)y(b)-x(a)y(a)] = \lim _{n \to \infty} \sum ^{n-1} _{j=1} 
\mathrm{E} \,[x(t_{j+1})y(t_j)-x(t_j)y(t_{j-1})]= \\
& \qquad \lim _{n \to \infty} \sum ^{n-1} _{j=1} 
\mathrm{E} \Big[\big(x(t_{j+1})- x(t_j)\big) \frac{y(t_j) +y(t_{j-1})}{2} + \\
& \phantom{\qquad \lim _{n \to \infty} \sum ^{n-1} _{j=1} 
\mathrm{E} \,[(x(t_{j+1}) -} \frac{x(t_{j+1})+x(t_j)}{2} \big(y(t_j)-y(t_{j-1})\big)\Big]= \\
& \lim _{n \to \infty} \sum ^{n-1} _{j=1} 
\mathrm{E} \,[\mathrm{D} x(t_{j}) \cdot y(t_j)+x(t_j)\mathrm{D}_*y(t_{j})] \frac{b-a}{n} = \\
& \int ^b _a \mathrm{E}\,[\mathrm{D}x(t) \cdot y(t) + x(t) \mathrm{D}_* y(t) ]\,dt .
\end{split}
\]
~~\textrm{QED}. \\

Now let us assume that the past $\mathscr{P}_t$ and the future $\mathscr{F}_t$ are conditionally 
independent given the present $\mathscr{P}_{t} \cap \mathscr{F}_t$. That is, if $f$ is 
any $\mathscr{F}_t$-measurable function in $\mathscr{L}^1$ then $\mathrm{E} \{ f 
\,|\, \mathscr{P}_{t} \} = \mathrm{E} \{ f \, |\, \mathscr{P}_{t} \cap \mathscr{F}_t \}$, and if $f$ is 
any $\mathscr{P}_t$-measurable function in $\mathscr{L}^1$ then \hbox{$\mathrm{E} \{ f 
\, |\, \mathscr{F}_{t} \} = \mathrm{E} \{ f \, |\, \mathscr{P}_{t} \cap \mathscr{F}_t \}$}. If $x$ is a 
Markov process and $\mathscr{P}_t$ is generated by the $x(s)$ with $s \le t$, and $\mathscr{F}_t$~by
the $x(s)$ with $s \ge t$, this is certainly the case. However, the assumption is much weaker. It 
applies, for example, to the position $x(t)$ of the Ornstein-Uhlenbeck process. The reason is that the 
present $ \mathscr{P}_{t} \cap \mathscr{F}_t$ may not be generated by $x(t)$; for example, in the 
Ornstein-Uhlenbeck case $v(t)=dx(t) / dt$ is also $ \mathscr{P}_{t} \cap \mathscr{F}_t$-measurable.

With the above assumption on the $\mathscr{P}_t$ and $\mathscr{F}_t$, if $x$ is an (S1) process 
then $\mathrm{D}x(t)$ and $\mathrm{D}_*x(t)$ are $ \mathscr{P}_{t} \cap \mathscr{F}_t$-measurable, 
and we can form 
$\mathrm{D}\mathrm{D}_*x(t)$ and $\mathrm{D}_*\mathrm{D}x(t)$ if they exist. 
Assuming they exist, we define
\begin{equation}
a(t) = \frac 12 \mathrm{D}\mathrm{D}_* x(t) + \frac 12 \mathrm{D}_*\mathrm{D}x(t)
\end{equation}
and call it the \emph{mean second derivative} or \emph{mean acceleration}.

If $x$ is a sufficiently smooth function of $t$ then $a(t) = d^2 x(t) / dt^2$. This is also true of other 
possible candidates for the title of mean acceleration, 
such as $\mathrm{D}\mathrm{D}_*x(t), \mathrm{D}_*\mathrm{D}x(t), \mathrm{D}\mathrm{D}x(t), 
\mathrm{D}_*\mathrm{D}_*x(t)$, and 
$\frac 12 \mathrm{D}\mathrm{D} x(t) + \frac 12 \mathrm{D}_*\mathrm{D}_*x(t)$. 
Of these the first four distinguish 
between the two choices of direction for the time axis, and so can be discarded. To discuss the 
fifth possibility, consider the Gaussian Markov process $x(t)$ satisfying
\[
dx(t) =-\omega x(t)\,dt + \,dw(t),
\]
where $w$ is a Wiener process, in equilibrium (that is, with the invariant Gaussian measure as initial 
measure). Then
\begin{align*}
\mathrm{D}x(t) & =-\omega x(t), \\
\mathrm{D}_*x(t) & =\phantom- \omega x(t), \\
a(t) & =- \omega ^2 x(t),
\end{align*}
but
\[
\frac 12 \mathrm{D}\mathrm{D} x(t) + \frac 12 \mathrm{D}_*\mathrm{D}_*x(t) = \omega ^2 x(t) .
\]
This process is familiar to us: it is the position in the Smoluchowski description of the highly 
overdamped harmonic oscillator (or the velocity of a free particle in the Ornstein-Uhlenbeck 
theory). The characteristic feature of this process is its constant tendency to go towards the origin, no
matter which direction of time is taken. Our definition of mean acceleration, which gives 
$a(t) =-\omega ^2 x(t)$, is kinematically the appropriate definition.
\begin{center}
\textrm{\bf Reference}
\end{center}

The stochastic integral was invented by It\^o:
\smallskip

\noindent [27]. Kiyosi It\^o, ``On Stochastic Differential Equations'',
Memoirs of the American Mathematical Society, Number 4 (1951).
\smallskip

Doob gave a treatment based on martingales [15, \S6, pp.\ 436--451]. Our discussion of stochastic 
integrals, as well as most of the other material of this section, is based on Doob's book.

\newpage

\fancyhead[CO]{DYNAMICS OF STOCHASTIC MOTION}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Dynamics of stochastic motion}

\hspace{6mm}The fundamental law of non-relativistic dynamics is Newton's law $F=ma$: the 
force on a particle 
is the product of the particle's mass and the acceleration of the particle. This law is, of course, 
nothing but the definition of force. Most definitions are trivial---others are profound. Feynman [28] 
has analyzed the characteristics that make Newton's definition profound:

``It implies that if we study the mass times the acceleration and call the product the force, i.e., if we 
study the characteristics of force as a program of interest, then we shall find that forces have 
some simplicity; the law is a good program for analyzing nature, it is a suggestion that the forces 
will be simple.''

Now suppose that $x$ is a stochastic process representing the motion of a particle of mass $m$. 
Leaving unanalyzed the dynamical mechanism causing the random fluctuations, we can ask how 
to express the fact that there is an external force $F$ acting on the particle. We do this simply 
by setting
\[
F = ma
\]
where $a$ is the mean acceleration (Chapter 11).

For example, suppose that $x$ is the position in the Ornstein-Uhlenbeck theory of Brownian motion, 
and suppose that the external force is $F=-\operatorname{grad}V$ where $\exp (- VD/m \beta)$ 
is integrable. In equilibrium, the particle has probability density a normalization constant times 
$\exp (- VD/m \beta)$ and satisfies
\begin{align*}
dx(t) & = v(t)dt \\
dv(t) & =-\beta v(t) dt + K\big(x(t)\big)dt + dB(t),
\end{align*}
where $K= F/m =-\operatorname{grad} V/m$, and $B$ has variance parameter $2 \beta ^2 D$. Then
\begin{align*}
\mathrm{D}x(t) & = \mathrm{D}_* x(t) = v(t), \\
\mathrm{D}v(t) & =-\beta v(t) + K\big(x(t)\big), \\
\mathrm{D}_*v(t) & = \beta v(t) + K\big(x(t)\big), \\
a(t) & = K\big(x(t)\big) .
\end{align*}
Therefore the law $F=ma$ holds.

%~ \\[1.5cm]
\begin{center}
\textrm{\bf Reference}
\end{center}
[28]. Richard P. Feynman, Robert B. Leighton, and Matthew Sands, 
``The Feynman Lectures on Physics'', Addison-Wesley, Reading, Massachusetts, 1963.

\newpage

\fancyhead[CO]{KINEMATICS OF MARKOVIAN MOTION}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Kinematics of Markovian motion}

\hspace{6mm}At this point I shall cease making regularity assumptions explicit. 
Whenever we take the derivative 
of a function, the function is assumed to be differentiable. Whenever we take $\mathrm{D}$ of a stochastic 
process, it is assumed to exist. Whenever we consider the probability density of a random variable, 
it is assumed to exist. I do this not out of laziness but out of ignorance. The problem of finding convenient regularity assumptions for this discussion and later applications of it (Chapter 15) is a non-trivial 
problem.

Consider a Markov process $x$ on $\Bbb{R}^{\ell}$ of the form
\[
dx(t) = b\big(x(t),t) dt + dw(t),
\]
where $w$ is a Wiener process on $\Bbb{R}^{\ell}$ with diffusion coefficient $\nu$ (we write 
$\nu$ instead of $D$ to avoid confusion with mean forward derivatives). Here $b$ is a fixed 
smooth function on $\Bbb{R}^{\ell +1}$. The $w(t)-w(s)$ are independent of the $x(r)$ 
whenever $r \le s$ and $ r \le t$, so that
\[
\mathrm{D}x(t) = b\big(x(t),t\big) .
\]

A Markov process with time reversed is again a Markov process (see Doob [15, \S6, p.\ 83]), 
so we can define $b_*$ by
\[
\mathrm{D}_*x(t) = b_* \big(x(t),t\big)
\]
and $w_*$ by
\[
dx(t) = b_* \big(x(t),t\big)dt +dw_*(t) .
\]

Let $f$ be a smooth function on $\Bbb{R}^{\ell +1}$. Then
\begin{align*}
f\big(x(t + \Delta t &),t+ \Delta t\big)-f\big(x(t),t\big) = \\
& \frac{\partial f}{\partial t} \big(x(t),t\big) \Delta t + [x(t + \Delta t)-x(t)] \cdot \nabla f\big(x(t),t\big) \\
& +\frac 12 \sum _{i,j} [x_i(t + \Delta t)-x_i(t)][x_j(t + \Delta t)-x_j(t)] 
\frac{\partial ^2 f}{\partial x^i \partial x^j} \big(x(t),t\big) \\
& + {\rm o}(t),
\end{align*}
so that
\begin{equation}
\mathrm{D}f\big(x(t),t\big) = ( \frac{\partial }{\partial t} + b \cdot \nabla + \nu \Delta) f\big(x(t),t\big) .
\end{equation}
Let $\nu _*$ be the diffusion coefficient of $w_*$. (A priori, $\nu_*$ might depends on $x$ and $t$, 
but we shall see shortly that $\nu _* = \nu$.) Similarly, we find
\begin{equation}
\mathrm{D}_*f\big(x(t),t\big) = \left( \frac{\partial }{\partial t} + b_* \cdot \nabla-\nu _* \Delta\right) f\big(x(t),t\big) .
\end{equation}

If $f$ and $g$ have compact support in time, then Theorem 11.12 shows that
\[
\int ^{\infty}_{- \infty} \mathrm{E}\mathrm{D}f\big(x(t),t\big) \cdot g\big(x(t),t\big) dt = 
- \int ^{\infty}_{- \infty} \mathrm{E}f\big(x(t),t\big) \mathrm{D}_* g\big(x(t),t\big)\, dt;
\]
that is,
\begin{align*}
& \quad \int ^{\infty}_{- \infty} \int _{ \Bbb{R}^{\ell}} \left( \frac{\partial }{\partial t} +
b \cdot \nabla + \nu \Delta\right)f(x,t) \cdot g(x,t) \rho (x,t)\, dxdt = \\
&-\int ^{\infty}_{- \infty} \int _{ \Bbb{R}^{\ell}} f(x,t) \left( \frac{\partial }{\partial t} + 
b_* \cdot \nabla-\nu _* \Delta\right)g(x,t) \cdot \rho (x,t)\, dxdt .
\end{align*}
For $A$ a partial differential operator, let $A^{\dag}$ be its (Lagrange) adjoint with respect to
Lebesgue measure on $\Bbb{R}^{\ell +1}$ and let $A^*$ be its adjoint with respect to $\rho$ times 
Lebesgue measure. Then $\int (Af) g \rho$ is equal to both $\int f A^{\dag} (g \rho)$ and 
$\int f (A^* g) \rho$, so that
\[
A^* = \rho ^{-1} A^{\dag} \rho .
\]
Now
\[
\left( \frac{\partial }{\partial t} + b \cdot \nabla + \nu \Delta\right)^{\dag} = 
- \frac{\partial }{\partial t}-b \cdot \nabla-\operatorname{div} b + \nu \Delta,
\]
so that
\begin{align*}
\rho ^{-1} \bigg( \frac{\partial }{\partial t} + b \cdot & \nabla + \nu \Delta\bigg)^{\dag} \rho g = 
\rho ^{-1} \left(- \frac{\partial }{\partial t}-b \cdot \nabla-\operatorname{div} b + \nu \Delta\right) ( \rho g)= \\
&-\frac{\partial g}{\partial t} -\rho ^{-1} \frac{\partial \rho}{\partial t}g 
-b \cdot \nabla g-\rho ^{-1} b \cdot (\operatorname{grad} \rho)g-(\operatorname{div} b)g \\
& \quad + \rho ^{-1}\nu \big(( \Delta \rho)g + 2 \operatorname{grad} \rho \cdot \operatorname{grad} g 
+\rho \Delta g\big).
\end{align*}
Recall the Fokker-Planck equation
\begin{equation}
\frac{\partial \rho}{\partial t} = -\operatorname{div} (b \rho)+ \nu \Delta \rho .
\end{equation}
Using this we find
\[
- \rho ^{-1} \frac{\partial \rho}{\partial t} = \frac{\operatorname{div} (b \rho)}{\rho} 
- \nu \frac{\Delta \rho}{\rho}= \operatorname{div} b +b \cdot \frac{\operatorname{grad} \rho }{\rho} 
- \nu \frac{\Delta \rho}{\rho},
\]
so we get
\[
- \frac{\partial }{\partial t}- b_* \cdot \nabla + \nu _* \Delta = 
- \frac{\partial }{\partial t}- b \cdot \nabla +2 \nu \frac{\operatorname{grad} \rho }{\rho} \cdot 
\nabla + \nu \Delta .
\]
Therefore, $\nu _* = \nu$ and $b_* = b-2 \nu (\operatorname{grad} \rho) / \rho$. If we make the 
definition
\[
u = \frac{b-b_*}{2},
\]
we have
\[
u = \nu \frac{\operatorname{grad} \rho }{\rho} .
\]
We call $u$ the \emph{osmotic velocity} $\big($cf.\ Chapter 4, Eq.\ (6)$\big)$.

There is also a Fokker-Planck equation for time reversed:
\begin{equation}
\frac{\partial \rho}{\partial t} = -\operatorname{div} (b_* \rho)- \nu \Delta \rho .
\end{equation}
If we define
\[
v= \frac{b+b_*}{2},
\]
we have the equation of continuity
\[
\frac{\partial \rho}{\partial t} = -\operatorname{div} (v \rho),
\]
obtained by averaging (13.3) and (13.4). We call $v$ the \emph{current velocity}.

Now
\[
u = \nu \frac{\operatorname{grad} \rho }{\rho} = \nu \operatorname{grad} \log \rho .
\]
Therefore,
\begin{align*}
\frac{\partial u}{\partial t} = & \nu \operatorname{grad} \frac{\partial }{\partial t}\log \rho =
\nu \operatorname{grad} \frac{\frac{\partial \rho}{\partial t}}{\rho} = \\
& \nu \operatorname{grad} \frac{\big(-\operatorname{div} (v \rho)\big)}{\rho} = 
- \nu \operatorname{grad} \left(\operatorname{div}v+v \cdot \frac{\operatorname{grad} \rho }{\rho}\right) = \\
&-\nu \operatorname{grad} \operatorname{div}v-\operatorname{grad}v \cdot u .
\end{align*}
That is,
\begin{equation}
\frac{\partial u}{\partial t} =-\nu \operatorname{grad} \operatorname{div}v -
\operatorname{grad}v \cdot u .
\end{equation}

Finally, from (13.1) and (13.2),
\begin{align*}
\mathrm{D}b_* \big(x(t),t\big) & = \frac{\partial }{\partial t} b_*\big(x(t),t\big) + b \cdot \nabla b_*\big(x(t),t\big) +
\nu \Delta b_*\big(x(t),t\big), \\
\mathrm{D}_*b \big(x(t),t\big) & = \frac{\partial }{\partial t} b\big(x(t),t\big) + b_* \cdot \nabla b\big(x(t),t\big)-
\nu \Delta b\big(x(t),t\big),
\end{align*}
so that the mean acceleration $\big($as defined in Chapter 11, Eq.\ (11.15)$\big)$ 
is given by $a\big(x(t),t\big)$ where
\[
a=\frac{\partial }{\partial t} \left(\frac{b+b_*}{2}\right)+\frac 12 b \cdot \nabla b_* +
\frac 12 b_* \cdot \nabla b-\nu \Delta \left(\frac{b-b_*}{2}\right) .
\]
That is,
\begin{equation}
\frac{\partial v}{\partial t} =a + u \cdot \nabla u-v \cdot \nabla v + \nu \Delta u .
\end{equation}

\newpage

\fancyhead[CO]{REMARKS ON QUANTUM MECHANICS}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Remarks on quantum mechanics}

\hspace{6mm}In discussing physical theories of Brownian motion we have seen
that physics has interesting ideas and 
problems to contribute to probability theory. Probabilities also play a fundamental r\^ole in quantum 
mechanics, but the notion of probability enters in a new way that is foreign both to classical 
mechanics and to mathematical probability theory. A mathematician interested in probability theory 
should become familiar with the peculiar concept of probability in quantum mechanics.

We shall discuss quantum mechanics from the point of view of the r\^ole of probabilistic concepts 
in it, limiting ourselves to the non-relativistic quantum mechanics of systems of finitely many degrees 
of freedom. This theory was discovered in 1925--1926. Its principal features were established quickly, 
and it has changed very little in the last forty years.

Quantum mechanics originated in an attempt to solve two puzzles: the discrete atomic spectra and 
the dual wave-particle nature of matter and radiation. Spectroscopic data were interpreted as being 
evidence for the fact that atoms are mechanical systems that can exist in stationary states only 
for a certain discrete set of energies.

There have been many discussions of the two-slit thought experiment illustrating the dual nature of 
matter; e.g., [28, \S12] and [29, Ch.\ 1]. Here we merely recall the bare facts: A particle issues from $\times$ in 
the figure, passes through the doubly-slitted screen in the middle, and hits the screen on the right, 
where its position is recorded. Particle arrivals are sharply localized indivisible events, but despite 
this the probability of arrival shows a complicated diffraction pattern typical of wave motion. If one of the 
holes is closed, there is no interference pattern. If an observation is made (using strong light of short 
wave length) to see which of the two slits the particle went through, there is again no interference pattern.

$$\vbox{\pdfimage{3.png}}$$
$$\hbox{Figure 3}$$
\bigskip

The founders of quantum mechanics can be divided into two groups: the reactionaries (Planck, Einstein, de 
Broglie, Schr\"odinger) and the radicals (Bohr, Heisenberg, Born, Jordan, Dirac). Correspondingly, quantum 
mechanics was discovered in two apparently different forms: wave mechanics and matrix mechanics. 
(Heisenberg's original term was ``quantum mechanics,'' and ``matrix mechanics'' is used when one wishes to 
distinguish it from Schr\"odinger's wave mechanics.)

In 1900 Planck introduced the quantum of action $h$ and in 1905 Einstein postulated particles of light with 
energy $E=h \nu$ ($\nu$ the frequency). We give no details as we shall not discuss radiation. In 1924, 
while a graduate student at the Sorbonne, Louis de Broglie put the two formulas $E=mc^2$ and 
$E=h \nu$ together and invented matter waves. The wave nature of matter received experimental 
confirmation in the Davisson-Germer electron diffraction experiment of 1927, and theoretical support 
by the work of Schr\"odinger in 1926. De Broglie's thesis committee included Perrin, Langevin, and Elie Cartan. 
Perhaps Einstein heard of de Broglie's work from Langevin. In any case, Einstein told Born, ``Read it; 
even though it looks crazy it's solid,'' and he published comments on de Broglie's work which Schr\"odinger read.

Suppose, with Schr\"odinger, that we have a particle (say an electron) of mass $m$ in a potential $V$. Here $V$ 
is a real function on $\Bbb{R}^3$ representing the potential energy. Schr\"odinger attempted to describe the 
motion of the electron by means of a quantity $\psi$ subject to a wave equation. He was led to the 
hypothesis that a stationary state vibrates according to the equation
\begin{equation}
\frac{\hbar ^2}{2m} \Delta \psi + (E-V) \psi =0,
\end{equation}
where $\hbar$ is Planck's constant $h$ divided by $2\pi$, and $E$ (with the dimensions of energy) plays 
the r\^ole of an eigenvalue.

This equation is similar to the wave equation for a vibrating elastic fluid contained in a given 
enclosure, except that $V$ is not a constant. Schr\"odinger was struck by another difference [30, p.\ 12]:

``A simplification in the problem of the `mechanical' waves (as compared with the fluid problem) 
consists in the absence of boundary conditions. I thought the latter simplification fatal when I first 
attacked these questions. Being insufficiently versed in mathematics, I could not imagine how proper 
vibration frequencies could appear \emph{without} boundary conditions.''

Despite these misgivings, Schr\"odinger found the eigenvalues and eigenfunctions of (14.1) for the case of the 
hydrogen atom, $V=- {\rm e} /r$ where ${\rm e}$ is the charge of the electron (and $- {\rm e}$ is the charge 
of the nucleus) and $r^2 = x^2 + y^2 + z^2$. The eigenvalues corresponded precisely to the known discrete 
energy levels of the hydrogen atom.

This initial triumph, in which discrete energy levels appeared for the first time in a natural way, 
was quickly followed by many others. Before the year of 1926 was out, Schr\"odinger reprinted six papers on 
wave mechanics in book form [30]. A young lady friend remarked to him (see the preface to [30]): ``When 
you began this work you had no idea that anything so clever would come out of it, had you?'' With this 
remark Schr\"odinger ``wholeheartedly agreed (with due qualification of the flattering adjective).''

Shortly before Schr\"odinger made his discovery, the matrix mechanics of Heisenberg appeared. In this 
theory one constructs six infinite matrices $q_k$ and $p_j$ ($j,k=1,2,3$) satisfying the commutation relations 
\[
p_j q_k -q_k p_j = \frac{\hbar}{i} \delta _{jk}
\]
and diagonalizes the matrix $H = p^2 /2m +V(q)$. Schr\"odinger remarks [30, p.\ 46]:

``My theory was inspired by L. de Broglie, Ann.\ de Physique (10)~3, p.\ 22, 1925 (Theses, Paris, 1924), 
and by brief, yet infinitely far-seeing remarks of A. Einstein, Berl.\ Ber., 1925, p.\ 9 et seq. I did not 
at all suspect any relation to Heisenberg's theory at the beginning. I naturally knew about his theory, 
but was discouraged, if not repelled, by what appeared to me as very difficult methods of 
transcendental algebra, and by the want of perspicuity (Anschaulichkeit).''

The remarkable thing was that where the two theories disagreed with the old quantum theory of Bohr, 
they agreed with each other (and with experiment!). Schr\"odinger quickly discovered the mathematical 
equivalence of the two theories, based on letting $q_k$ correspond to the operator of multiplication 
by the coordinate function $x_k$ and letting $p_j$ correspond to the operator $(\hbar /i) \partial / 
\partial x_j$ (see the fourth paper in [30]).

Schr\"odinger maintained (and most physicists agree) that the mathematical equivalence of two physical 
theories is not the same as their physical equivalence, and went on to describe a possible physical 
interpretation of the wave function $\psi$. According to this interpretation an electron with wave 
function $\psi$ is not a localized particle but a smeared out distribution of electricity with charge 
density $ {\rm e} \rho $ and electric current $ {\rm e} j$, where
\[
\rho = |\psi |^2, \qquad j = \frac{i \hbar}{2m} ( \psi \operatorname{grad} \bar{ \psi} 
- \bar{ \psi} \operatorname{grad} \psi) .
\]
(The quantities $\rho$ and $j$ determine $\psi$ except for a multiplicative factor of absolute value 
one.) This interpretation works very well for a single electron bound in an atom, provided one neglects 
the self-repulsion of the smeared out electron. However, when there are $n$ electrons, $\psi$ is a 
function on configuration space $\Bbb{R}^{3n}$, rather than coordinate space $\Bbb{R}^{3}$, 
which makes the interpretation of $\psi$ as a physically real object very difficult. Also, for free 
electrons $\psi$, and consequently $\rho$, spreads out more and more as time goes on; yet the arrival 
of electrons at a scintillation screen is always signaled by a sharply localized flash, never by a weak, 
spread out flash. These objections were made to Schr\"odinger's theory 
when he lectured on it in Copenhagen, and he reputedly said he wished he had never invented the theory.

The accepted interpretation of the wave function $\psi$ was put forward by Born [31], and quantum 
mechanics was given its present form by Dirac [32] and von Neumann [33]. Let us briefly describe quantum 
mechanics, neglecting superselection rules.

To each physical system there corresponds a Hilbert space $\mathscr{H}$. To every state (also called 
pure state) of the system there corresponds an equivalence class of unit vectors in $\mathscr{H}$, 
where $\psi _1$ and $\psi _2$ are called equivalent if $\psi _1 = a \psi _2$ for some complex number 
$a$ of absolute value one. (Such an equivalence class, which is a circle, is frequently called a ray.) 
The correspondence between states and rays is one-to-one. To each observable of the system there 
corresponds a self-adjoint operator, and the correspondence is again one-to-one. The development of the 
system in time is described by a family of unitary operators $U(t)$ on $\mathscr{H}$. There are two 
ways of thinking about this. In the Schr\"odinger picture, the state of the system changes with time---$\psi (t) = U(t) \psi _0$, where $\psi _0$ is the state at time~0, and observables do not change with 
time. In the Heisenberg picture, observables change with time---$A(t) = U(t)^{-1}A_0U(t)$, and the 
state does not change with time. The two pictures are equivalent, and it is a matter of convention 
which is used. For an isolated physical system, the dynamics is given by $U(t) = \exp (- (i / \hbar) 
Ht)$, where $H$, the Hamiltonian, is the self-adjoint operator representing the energy of the system.

It may happen that one does not know the state of the physical system, but merely that it is in state 
$\psi _1$ with probability $w_1$, state $\psi _2$ with probability $w_2$, etc., where $w_1 + 
w_2 + \ldots =1$. This is called a mixture (impure state), and we shall not describe its 
mathematical representation further.

The important new notion is that of a superposition of states. Suppose that we have two states 
$\psi _1$ and $\psi _2$. The number $| ( \psi _1, \psi _2)| ^2$ does not depend on the choice of 
representatives of the rays and lies between 0 and 1. Therefore, it can be regarded as a probability. If we 
know that the system is in the state $\psi _1$ and we perform an experiment to see whether or not 
the system is in the state $\psi _2$, then $| ( \psi _1, \psi _2)| ^2$ is the probability of finding that the 
system is indeed in the state $\psi _2$. We can write
\[
\psi _1 = ( \psi _2, \psi _1) \psi _2 + ( \psi _3, \psi _1) \psi _3
\]
where $\psi _3$ is orthogonal to $\psi _2$. We say that $\psi _1$ is a superposition of the states 
$\psi _2$ and $\psi _3$. Consider the mixture that is in the state $\psi _2$ with probability $| ( \psi _2, 
\psi _1)| ^2$ and in the state $\psi _3$ with probability $| ( \psi _3, \psi _1)| ^2$. Then $\psi _1$ and the 
mixture have equal probabilities of being found in the states $\psi _2$ and $\psi _3$, but they are 
quite different. For example, $\psi _1$ has the probability $| ( \psi _1, \psi _1)| ^2= 1$ of being found in the 
state $\psi _1$, whereas the mixture has only the probability $| ( \psi _2, \psi _1)| ^4 + | ( \psi _3, 
\psi _1)| ^4$ of being found in the state $\psi _1$.

A superposition represents a number of different possibilities, but unlike a mixture the different 
possibilities can interfere. Thus in the two-slit experiment, the particle is in a superposition of 
states of passing through the top slit and the bottom slit, and the interference of these possibilities 
leads to the diffraction pattern. If we look to see which slit the particle comes through then the 
particle will be in a mixture of states of passing through the top slit and the bottom slit and there 
will be no diffraction pattern.

If the system is in the state $\psi$ and $A$ is an observable with spectral projections $E_{\lambda}$ 
then $( \psi, E_{\lambda} \psi)$ is the probability that if we perform an experiment to determine the value 
of $A$ we will obtain a result $\le \lambda$. Thus $( \psi, A \psi) = \int \lambda ( \psi, 
d E_{\lambda} \psi)$ is the expected value of $A$ in the state $\psi$. (The left hand side is 
meaningful if $\psi$ is in the domain of $A$; the integral on the right hand side converges if $\psi$ is 
merely in the domain of $|A|^{\frac 12}$.) The observable $A$ has the value $\lambda$ with certainty if 
and only if $\psi$ is an eigenvector of $A$ with eigenvalue $\lambda$, $A \psi = \lambda \psi$.

Thus quantum mechanics differs from classical mechanics in not requiring every observable to have a 
sharp value in every (pure) state. Furthermore, it is in general impossible to find a state such that two given 
observables have sharp values. Consider the position operator $q$ and the momentum operator $p$ for 
a particle with one degree of freedom, and let $\psi$ be in the domain of $p^2, q^2,pq$, and $qp$. Then 
$( \psi, p^2 \psi)-( \psi, p \psi)^2 = ( \psi, \big(p-( \psi, p \psi)\big)^2 \psi)$ is the variance of the 
observable $p$ in the state $\psi$ and its square root is the standard deviation, which physicists 
frequently call the dispersion and denote by $ \Delta p$. Similarly for $( \psi, q^2 \psi)-
( \psi, q \psi)^2$. We find, using the commutation rule
\begin{equation}
(pq-qp) \psi = \frac{\hbar}{i} \psi,
\end{equation}
%%% we find that for all real $\alpha$,
\begin{align*}
0 \le \big(( \alpha q & + ip) \psi, ( \alpha q + ip) \psi\big) = \\
& \alpha ^2( \psi, q^2 \psi)-i \alpha \big( \psi, (pq -qp) \psi\big) + ( \psi, p^2 \psi) = \\
& \alpha ^2( \psi, q^2 \psi)-\alpha \hbar + ( \psi, p^2 \psi) .
\end{align*}
Since this is positive for all real $\alpha$, the discriminant must be negative,
\begin{equation}
\hbar ^2-4 ( \psi, q^2 \psi) ( \psi, p^2 \psi) \le 0 .
\end{equation}
The commutation relation (14.2) continues to hold if we replace $p$ by $p-( \psi, p \psi)$ and $q$ 
by $q-( \psi, q \psi)$, so (14.3) continues to hold after this replacement. That is,
\begin{equation}
\Delta q \Delta p \ge \frac{\hbar}{2} .
\end{equation}
This is the well-known proof of the Heisenberg uncertainty relation. The great importance of 
Heisenberg's discovery, however, was not the formal deduction of this relation but the presentation 
of arguments that showed, in an endless string of cases, that the relation (14.4) must hold on 
physical grounds independently of the formalism.

Thus probabilistic notions are central in quantum mechanics. Given the state $\psi$, the observable $A$ can be 
regarded as a random variable on the probability space consisting of the real line with the measure $( \psi, 
d E_{\lambda} \psi)$, where the $E_{\lambda}$ are the spectral projections of $A$. Similarly, any 
number of commuting self-adjoint operators can be regarded as random variables on a probability space. 
(Two self-adjoint operators are said to commute if their spectral projections commute.) But, and it 
is this which makes quantum mechanics so radically different from classical theories, the set of all 
observables of the system in a given state cannot be regarded as a set of random variables on a probability 
space. For example, the formalism of quantum mechanics does not allow the possibility of $p$ and $q$ 
both having sharp values even if the putative sharp values are unknown.

For a while it was thought by some that there might be ``hidden variables''---that is, a more refined 
description of the state of a system---which would allow all observables to have sharp values if a 
complete description of the system were known. Von Neumann [33] showed, however, that any such 
theory would be a departure from quantum mechanics rather than an extension of it. It follows from von 
Neumann's theorem that the set of all self-adjoint operators in a given state cannot be regarded as a family 
of random variables on a probability space. Here is another result along these lines.
\\

%\begin{thm}
T{\ninerm HEOREM} 14.1 {\it
Let $A=(A_1, \ldots, A_n)$ be an $n$-tuple of operators on a Hilbert space $\mathscr{H}$ such that 
for all $x$ in $\Bbb{R}^n$,
\[
x \cdot A = x_1 A_1 + \ldots + x_n A_n
\]
is essentially self-adjoint. Then either the $(A_1, \ldots, A_n)$ commute or there is a $\psi$ in 
$\mathscr{H}$ with $ \| \psi \| =1$ such that there do not exist random variables $\alpha _1, \ldots, 
\alpha _n$ on a probability space with the property that for all $x$ in $\Bbb{R}^n$ and $\lambda$ in 
$\Bbb{R}$,
\[
\mathrm{Pr} \{x \cdot \alpha \le \lambda \} = \big( \psi, E_{\lambda} (x \cdot A) \psi\big)
\]
where $x \cdot \alpha = x_1 \alpha _1 + \ldots +x_n \alpha _n$ and the $E_{\lambda} (x \cdot A)$ 
are the spectral projections of the closure of $x \cdot A$.
}
\\
%\end{thm}

In other words, $n$ observables can be regarded as random variables, in all states, if and only if 
they commute. \\

{\it Proof.} We shall not distinguish notationally between $x \cdot A$ and its closure.

Suppose that for each unit vector $\psi$ in $\mathscr{H}$ there is such an $n$-tuple $\alpha$ of 
random variables, and let $\mu _{ \psi}$ be the probability distribution of $\alpha$ on~$\Bbb{R}^n$. 
That is, for each Borel set $B$ in $\Bbb{R}^n$, $\mu _{ \psi} (B) = \mathrm{Pr} \{ \alpha \in B \}$. 
If we integrate first over the hyperplanes orthogonal to $x$, we find that
\begin{align*}
\int _{\Bbb{R}^n} e^{i x \cdot \xi} d \mu _{ \psi} ( \xi) & = \int ^{\infty}_{- \infty} e^{i \lambda} 
d\, \mathrm{Pr} \{x \cdot \alpha \le \lambda \} \\
& = \int ^{\infty}_{- \infty} e^{i \lambda} \big( \psi, d E_{\lambda} (x \cdot A) \psi\big) = 
( \psi, e^{i x \cdot A} \psi) .
\end{align*}
Thus the measure $\mu _{ \psi}$ is the Fourier transform of $( \psi, e^{i x \cdot A} \psi)$. By the 
polarization identity, if $\varphi$ and $\psi$ are in $\mathscr{H}$ there is a complex measure 
$\mu _{ \varphi \psi}$ such that $\mu _{ \varphi \psi}$ is the Fourier transform of $( \varphi, 
e^{i x \cdot A} \psi)$ and $\mu _{ \psi \psi} = \mu _{ \psi}$. For any Borel set $B$ in $\Bbb{R}^n$ 
there is a unique operator $\mu (B)$ such that $\big( \varphi, \mu (B) \psi\big) = \mu _{ \varphi \psi} (B)$, 
since $\mu _{ \varphi \psi}$ depends linearly on $\psi$ and antilinearly on $\psi$. Thus we have
\[
\int _{\Bbb{R}^n} e^{i x \cdot \xi} \big( \varphi, d \mu ( \xi) \psi\big)=( \varphi, e^{i x \cdot A} \psi) .
\]

The operator $\mu (B)$ is positive since $\mu _{ \psi}$ is a positive measure. Consequently, if we 
have a finite set of elements $\psi _j$ of $\mathscr{H}$ and corresponding points $x_j$ of 
$\Bbb{R}^n$, then
\begin{align*}
& \sum _{j,k} \big( \psi _k, e^{i(x_j-x_k) \cdot A} \psi _j\big) = \\
& \sum _{j,k} \int _{\Bbb{R}^n} e^{i(x_j-x_k) \cdot \xi } \big( \psi _k, d \mu ( \xi) \psi _j\big)= \\
& \int _{\Bbb{R}^n} \big( \psi ( \xi), d \mu ( \xi) \psi \big( \xi)\big) \ge 0,
\end{align*}
where
\[
\psi ( \xi) = \sum _{j} e^{i x_j \cdot \xi } \psi _j .
\]
Furthermore, $e^{i 0 \cdot A} =1$ and $e^{i (-x) \cdot A}=(e^{ix \cdot A})^*$. Under these conditions, the 
theorem on unitary dilations of Nagy [34, Appendix, p.\ 21] implies that there is a Hilbert space 
$\mathscr{K}$ containing $\mathscr{H}$ and a unitary representation $x \mapsto U(x)$ of $\Bbb{R}^n$ on 
$\mathscr{K}$ such that, if $E$ is the orthogonal projection of $\mathscr{K}$ onto $\mathscr{H}$, then
\[
EU(x) \psi = e^{ix \cdot A} \psi
\]
for all $x$ in $\Bbb{R}^n$ and all $\psi$ in $\mathscr{H}$. Since $e^{i x \cdot A}$ is already unitary, 
\[
\| U(x) \psi \| = \| e^{ix \cdot A} \psi \| = \| \psi \|,
\]
so that $\| EU(x) \psi \| = \| U(x) \psi \|$. Consequently, $EU(x) \psi = U(x) \psi $ and each $U(x)$ maps 
$\mathscr{H}$ into itself, so that $U(x) \psi = e^{ix \cdot A} \psi $ for all $\psi$ in $\mathscr{H}$. 
Since $x \mapsto U(x)$ is a unitary representation of the commutative group~$\Bbb{R}^n$, the $e^{ix 
\cdot A}$ all commute, and consequently the $A_j$ commute. ~~\textrm{QED}. \\

Quantum mechanics forced a major change in the notion of reality. The position and momentum of a 
particle could no longer be thought of as properties of the particle. They had no real existence before 
measurement of the one with a given accuracy precluded the measurement of the other with too great 
an accuracy, in accordance with the uncertainty principle. This point of view was elaborated by Bohr 
under the slogan of ``complementarity'', and Heisenberg wrote a book [35] explaining the physical basis 
of the new theory.

At the Solvay Congress in 1927, Einstein was very quiet, but when pressed objected that $\psi$ could 
not be the complete description of the state. For example, the wave function in Fig.\ 4 would have 
axial symmetry, but the place of arrival of an individual particle on the hemispherical screen does not 
have this symmetry. The answer of quantum mechanics is that the symmetrical wave function $\psi$ 
describes the state of the system before a measurement is made, but the act of measurement changes $\psi$.

$$\vbox{\pdfimage{4.png}}$$
$$\hbox{Figure 4}$$
\bigskip

To understand the r\^ole of probability in quantum mechanics it is necessary to discuss measurement. 
The quantum theory of measurement was created by von Neumann [33]. A very readable summary is the 
book by London and Bauer [36]. See also two papers of Wigner [37] [38], which we follow now.

Consider a physical system with wave function $\psi$ in a state of superposition of the two 
orthogonal states $\psi _1$ and $\psi _2$, so that $\psi = \alpha _1 \psi _1 +\alpha _2 \psi _2 $. 
We want to perform an experiment to determine whether it is in the state $\psi _1$ or the state 
$\psi _2$. (We know that the probabilities are respectively $| \alpha _1 |^2$ and $| \alpha _2 |^2$, 
but we want to know which it is in.)

If $A$ is any observable, the expected value of $A$ is
\begin{multline*}
\big( \alpha _1 \psi _1 +\alpha _2 \psi _2, A( \alpha _1 \psi _1 +\alpha _2 \psi _2)\big) = \\
| \alpha _1 |^2 ( \psi _1, A\psi _1) + | \alpha _2 |^2 ( \psi _2, A \psi _2)+
\bar{ \alpha}_1 \alpha _2 ( \psi _1, A\psi _2) + \bar{ \alpha}_2 \alpha _1 ( \psi _2, A\psi _1) .
\end{multline*}
Suppose now we couple the system to an apparatus designed to measure whether the system is in the 
state $\psi _1$ or $\psi _2$, and that after the interaction the system plus apparatus is in the state
\[
\varphi = \alpha _1 \psi _1 \otimes \chi _1 +\alpha _2 \psi _2 \otimes \chi _2,
\]
where $\chi _1$ and $\chi _2$ are orthogonal states of the apparatus. If $A$ is any observable 
pertaining to the system alone, then
\[
( \varphi, A \otimes 1 \varphi) = 
| \alpha _1 |^2 ( \psi _1, A\psi _1) + | \alpha _2 |^2 ( \psi _2, A \psi _2) .
\]
Thus, after the interaction with the apparatus, the system behaves like a mixture of $\psi _1$ and 
$\psi _2$ rather than a superposition. It is a measurement of whether the system is in the state 
$\psi _1$ or $\psi _2$.

However, observe that knowing the state of the system plus apparatus after the interaction tells us 
nothing about which state the system is in! The state $\varphi$ is a complete description of the 
system plus apparatus. It is causally determined by $\psi$ and the initial state of the apparatus, 
according to the Schr\"odinger equation governing the interaction. 
Thus letting the system interact with an apparatus can never give us more information.

If we knew that after the interaction the apparatus is in the state $\chi _1$ we would know that the 
system is in the state $\psi _1$. But how do we tell whether the apparatus is in state $\chi _1$ or 
$\chi _2$? We might couple it to another apparatus, but this threatens an infinite regress.

In practice, however, the apparatus is macroscopic, like a spot on a photographic plate or the pointer 
on a meter, and I merely look to see which state it is in, $\chi _1$ or $\chi _2$. After I have become 
aware of the state $\chi _1$ or $\chi _2$, the act of measurement is complete. If I see that the 
apparatus is in state $\chi _1$, the system plus apparatus is in state $\psi _1 \otimes \chi _1$ and 
the system is in state $\psi _1$. 
(As we already knew, this will happen with probability $| \alpha _1 |^2$.) 
Similarly for $\chi _2$. After the interaction but before awareness has dawned in me, the state of 
the system plus apparatus is $\alpha_1\psi_1\otimes\chi_1+\alpha_2\psi_2\otimes\chi_2$;
the instant I become aware of the state of the apparatus, the state of the system plus apparatus is
either $\psi _1 \otimes \chi _1$ or $\psi _2 \otimes \chi _2$. Thus the 
state can change in two ways: continuously, linearly, and causally by means of the Schr\"odinger equation 
or abruptly, nonlinearly, and probabilistically by means of my consciousness. The latter change is 
called ``reduction of the wave packet''. Concerning the reduction of the wave packet, Wigner [39] writes:

``This takes place whenever the result of an observation enters the consciousness of the observer---or, 
to be even more painfully precise, my own consciousness, since I am the only observer, all other 
people being only subjects of my observations.''

This theory is known as the orthodox theory of measurement in quantum mechanics. The word ``orthodox'' is 
well chosen: one suspects that many practicing physicists are not entirely orthodox in their beliefs.

Of those founders of quantum mechanics whom we labeled ``reactionaries'', none permanently accepted the 
orthodox interpretation of quantum mechanics. Schr\"odinger writes [40, p.\ 16]:

``For it must have given to de Broglie the same shock and disappointment as it gave to me, when 
we learnt that a sort of transcendental, almost psychical interpretation of the wave phenomenon had 
been put forward, which was very soon hailed by the majority of leading theorists as the only one 
reconcilable with experiments, and which has now become the orthodox creed, accepted by almost 
everybody, with a few notable exceptions.''

The literature on the interpretation of quantum mechanics contains much of interest, but I shall discuss only 
three memorable paradoxes: the paradox of Schr\"odinger's cat [41, p.\ 812], the paradox of the nervous 
student [42] [43] [44], and the paradox of Wigner's friend [38]. The original accounts of these 
paradoxes make very lively reading.

One is inclined to accept rather abstruse descriptions of electrons and atoms, which one has never 
seen. Consider, however, a cat that is enclosed in a vault with the following infernal machine, 
located out of reach of the cat. A small amount of a radioactive substance is present, with a half-life 
such that the probability of a single decay in an hour is about one half. If a radioactive decay occurs, a 
counter activates a device that breaks a phial of prussic acid, killing the cat. The only point of this 
paradox is to consider what, according to quantum mechanics, is a \emph{complete description} of the 
state of affairs in the vault at the end of an hour. One cannot say that the cat is alive or dead, but that 
the state of it and the infernal machine is described by a superposition of various states containing 
dead and alive cats, in which the cat variables are mixed up with the machine variables. This precise 
state of affairs is the ineluctable outcome of the initial conditions. Unlike most thought experiments, 
this one could actually be performed, were it not inhumane.

The first explanations of the uncertainty principle (see [35]) made it seem the natural result of the 
fact that, for example, observing the position of a particle involves giving the particle a kick and thereby 
changing its momentum. Einstein, Podolsky, and Rosen [42] showed that the situation is not that 
simple. Consider two particles with one degree of freedom. Let $x_1$ and $p_1$ denote the position 
and momentum operators of the first, $x_2$ and $p_2$ those of the second. Now $x=x_1-x_2$ and 
$p=p_1+p_2$ are commuting operators and so can simultaneously have sharp values, say $x'$ and $p'$ 
respectively. Suppose that $x'$ is very large, so that the two particles are very far apart. Then we 
can measure $x_2$, obtaining the value $x'_2$, say, without in any way affecting the first particle. 
A measurement of $x_1$ then must give $x'+x'_2$. Since the measurement of $x_2$ cannot have 
affected the first particle (which is very far away), there must have been something about the 
condition of the value $x'_1 =x'+x'_2$ which meant that $x_1$ if
measured would give the value $x_1'=x'+x_2'$. Similarly for position measurements. 
To quote Schr\"odinger [44]:

``Yet since I can predict \emph{either} $x'_1$ \emph{or} $p'_1$ without interfering with system No.\ 1 
and since system No.\ 1, like a scholar in an examination, cannot possibly know which of the two 
questions I am going to ask it first: it so seems that our scholar is prepared to give the right answer 
to the \emph{first} question he is asked, \emph{anyhow}. Therefore he must know both answers; which 
is an amazing knowledge, quite irrespective of the fact that after having given his first answer our 
scholar is invariably so disconcerted or tired out, that all the following answers `wrong'.''

The paradox of Wigner's friend must be told in the first person. There is a physical system in the  
state~$\psi = \alpha _1 \psi _1 +\alpha _2 \psi _2 $ which, if in 
the state~$\psi _1$, produces a flash, and 
if in the state $\psi _2$, does not. In the description of the measurement process given earlier, for 
the apparatus I substitute my friend. After the interaction of system and friend they are in the state 
$\varphi = \alpha _1 \psi _1 \otimes \chi _1 +\alpha _2 \psi _2 \otimes \chi _2$. I ask my friend if 
he saw a flash. If I receive the answer ``yes'' the state changes abruptly (reduction of the wave packet) 
to $\psi _1 \otimes \chi _1$; if I receive the answer ``no'' the state changes to $\psi _2 \otimes 
\chi _2$. Now suppose I ask my friend, ``What did you feel about the flash before I asked you?'' 
He will answer that he already knew that there was (or was not) a flash before I asked him. If I 
accept this, I must accept that the state was $\psi _1 \otimes \chi _1$ (or $\psi _2 \otimes 
\chi _2$), rather than $\varphi$, in violation of the laws of quantum mechanics. One possibility is to deny 
the existence of consciousness in my friend (solipsism). Wigner prefers to believe that the laws of 
quantum mechanics are violated by his friend's consciousness.

%~ \\[1cm]
\begin{center}
\textrm{\bf References}
\end{center}
[29]. R. P. Feynman and A. R. Hibbs, ``Quantum Mechanics and Path Integrals'', McGraw-Hill, New York, 1965.

\smallskip

\noindent [30]. Erwin Schr\"odinger, ``Collected Papers on Wave Mechanics'', 
translated by J. F. Shearer and W. M. Deans, 
Blackie \& Son limited, London, 1928.

\smallskip

\noindent [31]. Max Born, \emph{Zur Quantenmechanik der Stossvorg\"ange}, Zeitschrift f\"ur Physik 
\textrm{37} (1926), 863--867, \textrm{38}, 803--827.

\smallskip

\noindent [32]. P. A. M. Dirac, ``The Principles of Quantum Mechanics'', Oxford, 1930.

\smallskip

\noindent [33]. John von Neumann, ``Mathematical Foundations of Quantum Mechanics'', 
translated by R. T. Beyer, Princeton University Press, Princeton, 1955.

\smallskip

\noindent [34]. Frigyes Riesz and B\'ela Sz.-Nagy, ``Functional Analysis'', 
2nd Edition, translated by L. F. Boron, with Appendix: Extensions of Linear Transformations in Hilbert 
Space which Extend Beyond This Space, by B. \hbox{Sz.-Nagy}, 
Frederick Ungar Publishing Co., New York, 1960.

\smallskip

\noindent [35]. Werner Heisenberg, ``The Physical Principles of the Quantum Theory'', 
translated by Carl Eckhart and Frank C. Hoyt, Dover Publications, New York, 1930.

\smallskip

\noindent [36]. F. London and E. Bauer, ``La th\'eorie de l'observation en m\'ecanique quantique'', 
Hermann et Cie., Paris, 1939.

\smallskip

\noindent [37]. Eugene P. Wigner, \emph{The problem of measurement}, 
American Journal of Physics \textrm{31} (1963), 6--15.

\smallskip

\noindent [38]. Eugene P. Wigner, \emph{Remarks on the mind-body question}, 
pp.\ 284--302 in ``The Scientist Speculates'', edited by I. J. Good, 1962.

\smallskip

\noindent [39]. Eugene P. Wigner, \emph{Two kinds of reality}, The Monist \textrm{48} (1964), 248--264.

\smallskip

\noindent [40]. Erwin Schr\"odinger, \emph{The meaning of wave mechanics}, 
pp.\ 16--30 in ``Louis de Broglie physicien et penseur'', 
edited by Andr\'e George, \'editions Albin Michel, Paris, 1953.

\smallskip

\noindent [41]. Erwin Schr\"odinger, \emph{Die gegenw\"artige Situation in der Quantenmechanik}, 
published as a serial in Naturwissenschaften \textrm{23} (1935), 807--812, 823--828, 844--849.

\smallskip

\noindent [42]. A. Einstein, B. Podolsky and N. Rosen, 
\emph{Can quantum-mechanical description of reality be considered complete?}, 
Physical Review \textrm{47} (1935), 777--780.

\smallskip

\noindent [43]. N. Bohr, \emph{Can quantum-mechanical description of reality be considered complete?}, 
Physical Review \textrm{48} (1935), 696--702.

\smallskip

\noindent [44]. Erwin Schr\"odinger, \emph{Discussion of probability relations between separated systems}, 
Proceedings of the Cambridge Philosophical Society \textrm{31} (1935), 555-563.

\smallskip

An argument against hidden variables that is much more incisive than von Neumann's is 
presented in a forthcoming paper:

\smallskip

\noindent [45]. Simon Kochen and E. P. Specker, \emph{The problem of hidden variables in quantum mechanics}, 
Journal of Mathematics and Mechanics, \textrm{17} (1967), 59--87.

\newpage
\thispagestyle{empty}

\fancyhead[CO]{BROWNIAN MOTION IN THE AETHER}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Brownian motion in the aether}

\hspace{6mm}Let us try to see whether some of the puzzling physical phenomena that occur on the atomic scale 
can be explained by postulating a kind of Brownian motion that agitates all particles of matter, 
even particles far removed from all other matter. It is not necessary to think of a material model 
of the aether and to imagine the cause of the motion to be bombardment by grains of the aether. 
Let us, for the present, leave the cause of the motion unanalyzed and return to Robert Brown's 
conception of matter as composed of small particles that exhibit a rapid irregular motion having 
its origin in the particles themselves (rather like Mexican jumping beans).

We cannot suppose that the particles experience friction in moving through the aether as this 
would imply that uniform rectilinear motion could be distinguished from absolute rest. Consequently, 
we cannot base our discussion on the Langevin equation.

We shall assume that every particle performs a Markov process of the form
\begin{equation}
dx(t) = b\big(x(t),t\big)dt + dw(t),
\end{equation}
where $w$ is a Wiener process on $\Bbb{R}^3$, with $w(t)-w(s)$ independent of $x(r)$ whenever 
$r \le s \le t$. Macroscopic bodies do not appear to move like this, so we shall postulate that the 
diffusion coefficient $\nu$ is inversely proportional to the mass $m$ of the particle. We write it as 
\[
\nu = \frac{ \hbar}{2m} .
\]
The constant $ \hbar$ has the dimensions of action. If $ \hbar$ is of the order of Planck's constant 
$h$ then the effect of the Wiener process would indeed not be noticeable for macroscopic bodies but 
would be relevant on the atomic scale. (Later we will see that $ \hbar = h/2 \pi$.) The kinematical 
assumption~(15.1) is non-relativistic, and the theory we are proposing is meant only as an 
approximation valid when relativistic effects can safely be neglected.

We have already (Chapter 13) studied the kinematics of such a process. We let $b_*$ be the mean 
backward velocity, $u=(b-b_*)/2$, $v=(b+b_*)/2$. By (13.5) and (13.6) of Chapter 13,
\begin{equation}
\begin{split}
\frac{\partial u}{\partial t} & =-\frac{ \hbar}{2m} \operatorname{grad} \operatorname{div}v- 
\operatorname{grad}v \cdot u, \\
\frac{\partial v}{\partial t} 
& = a -v \cdot \nabla v + u \cdot \nabla u + \frac{ \hbar}{2m} \Delta u,
\end{split}
\end{equation}
where $a$ is the mean acceleration.

Suppose that the particle is subject to an external force $F$. We make the dynamical assumption 
$F=ma$, and substitute $F/m$ for $a$ in (15.2). $\big($This agrees with what is done in the 
Ornstein-Uhlenbeck theory of Brownian motion with friction (Chapter 12).$\big)$

Consider the case when the external force is derived from a potential~$V$, which may be 
time-dependent, so that $F(x,t) =-\operatorname{grad}V(x,t)$. Then (15.2) becomes
\begin{equation}
\begin{split}
\frac{\partial u}{\partial t} & =-\frac{\hbar}{2m} \operatorname{grad} \operatorname{div}v- 
\operatorname{grad}v \cdot u, \\
\frac{\partial v}{\partial t} 
& =-\frac{1}{m} \operatorname{grad}V -v \cdot \nabla v + u \cdot \nabla u + 
\frac{ \hbar}{2m} \Delta u .
\end{split}
\end{equation}
If $u_0(x)$ and $v_0(x)$ are given, we have an initial value problem: to solve the system (15.3) of 
coupled non-linear partial differential equations subject to $u(x,0) =u_0(x), v(x,0)=v_0(x)$ for all 
$x$ in $\Bbb{R}^3$. Notice that when we do this we are not solving the equations of motion of 
the particle. We are merely finding what stochastic process the particle obeys, with the given 
force and the given initial osmotic and current velocities. Once $u$ and $v$ are known, $b$, $b*$, 
and $\rho$ are known, and so the Markov process is known.

It would be interesting to know the general solution of the initial value problem (15.3). However, 
I can only solve it with the additional assumption that $v$ is a gradient. (We already know (Chapter 13) 
that $u$ is a gradient.) A solution of the problem without this assumption would seem to correspond 
to finding the Markov process of the particle when the containing fluid, the aether, is in motion.

Let $R=\frac 12 \log \rho$. Then we know (Chapter 13) that
\begin{equation}
\operatorname{grad}R= \frac{m}{ \hbar}u .
\end{equation}
Assuming that $v$ is also a gradient, let $S$ be such that
\begin{equation}
\operatorname{grad}S= \frac{m}{ \hbar}v .
\end{equation}
Then $S$ is determined, for each $t$, up to an additive constant.

It is remarkable that the change of dependent variable
\begin{equation}
\psi =e^{R+iS}
\end{equation}
transforms (15.3) into a linear partial differential equation; in fact, into the Schr\"odinger equation
\begin{equation}
\frac{\partial \psi}{\partial t}=i \frac{\hbar}{2m} \Delta \psi -i \frac{1}{\hbar}V \psi +
i \alpha (t) \psi .
\end{equation}
(Since the integral of $\rho = \bar{\psi} \psi$ is independent of $t$, if (15.7) holds at all then 
$\alpha (t)$ must be real. By choosing, for each $t$, the arbitrary constant in~$S$ appropriately 
we can arrange for $\alpha (t)$ to be 0.)

To prove (15.7), we compute the derivatives and divide by $\psi$, finding
\[
\frac{\partial R}{\partial t}
+i \frac{\partial S}{\partial t}= i \frac{\hbar}{2m} \big( \Delta R +i \Delta S +
[ \operatorname{grad} (R+iS)]^2\big)- i \frac{1}{\hbar} V +i \alpha (t) .
\]
Taking gradients and separating real and imaginary parts, we see that this is equivalent to the 
pair of equations
\begin{align*}
\frac{\partial u }{\partial t}
& =- \frac{\hbar}{2m} \Delta v- \operatorname{grad} v \cdot u, \\
\frac{\partial v}{\partial t} & = \frac{\hbar}{2m} \Delta u + \frac 12 \operatorname{grad} u^2
-\frac 12 \operatorname{grad} v^2- \frac{1}{m} \operatorname{grad} V .
\end{align*}
Since $u$ and $v$ are gradients, this is the same as (15.3).

Conversely, if $\psi$ satisfies the 
Schr\"odinger equation (15.7) and we define $R,S,u,v$ by (15.6), (15.4), and (15.5), then $u$ and 
$v$ satisfy (15.3). Note that $u$ becomes singular when $\psi =0$.

Is it an accident that Markov processes of the form (15.1) with the dynamical law $F=ma$ are 
formally equivalent to the Schr\"odinger equation? As a test, let us consider the motion of a 
particle in an external electromagnetic field. Let, as is customary, $A$ be the vector potential, 
$\varphi$~the scalar potential, $E$ the electric field strength, $H$ the magnetic field strength, 
and $c$ the speed of light. Then
\begin{equation}
\phantom{xxxx.}H = \operatorname{curl} A,
\end{equation}
\vskip-20pt
\begin{equation}
E+ \frac{1}{c} \frac{\partial A}{\partial t} =- \operatorname{grad} \varphi .
\end{equation}
The Lorentz force on a particle of charge $\mathrm{e}$ is
\begin{equation}
F= \mathrm{e} \left(E+ \frac{1}{c} V \times H\right)\!,
\end{equation}
where $v$ is the classical velocity. We adopt (15.10) as the force on a particle undergoing the 
Markov process (15.1) with $v$ the current velocity. We do this because the force should be 
invariant under time inversion $t \mapsto -t$, and indeed $H \mapsto -H$, $v \mapsto -v$ (while $u \mapsto u$) under 
time inversion. As before, we substitute $F/m$ for $a$ in (15.2). Now, however, we assume the 
generalized momentum $mv+ \mathrm{e} A/c$ to be a gradient. 
(This is a gauge invariant assumption.) Letting $\operatorname{grad} R=mu/ \hbar$ as before, 
we define $S$ up to an additive function of $t$ by
\[
\operatorname{grad} S=\frac{m}{\hbar} \left(v+ \frac{\mathrm{e}}{mc} A\right)\!,
\]
and let
\[
\psi =e^{R+iS}.
\]
Then $\psi$ satisfies the Schr\"odinger equation
\begin{equation}
\frac{\partial \psi}{\partial t}=- \frac{i}{2m \hbar} \left(-i \hbar \nabla-\frac{\mathrm{e}}{c}A\right)^2 \psi 
- \frac{i \mathrm{e}}{\hbar} \varphi \psi + i \alpha (t) \psi,
\end{equation}
where as before $\alpha (t)$ is real and can be made 0 by a suitable choice of $S$.

To prove (15.11), we perform the differentiations and divide by $\psi$, obtaining
\begin{align*}
& \qquad \frac{\partial R}{\partial t}+i \frac{\partial S}{\partial t}= 
i \frac{\hbar}{2m} \big( \Delta R+i \Delta S+ [ \operatorname{grad} (R+iS)]^2\big) \\
& + \frac{\mathrm{e}}{mc} A \cdot \operatorname{grad} (R+iS) + \frac 12 \frac{\mathrm{e}}{mc} 
\operatorname{div} A- \frac{i \mathrm{e}^2}{2m \hbar c^2} A^2-\frac{i \mathrm{e}}{\hbar} \varphi 
+i \alpha (t) .
\end{align*}
This is equivalent to the pair of equations we obtain by taking gradients and separating real and 
imaginary parts. For the real part we find
\begin{align*}
\frac{m}{\hbar} \frac{\partial u}{\partial t}
= &-\frac{\hbar}{2m} \operatorname{grad} \operatorname{div} (v+ \frac{\mathrm{e}}{mc}A) \\
&-\frac{\hbar}{2m} \operatorname{grad} \left[2 \frac{m}{\hbar} u \cdot \frac{m}{\hbar} 
\left(v+ \frac{\mathrm{e}}{mc} A\right)\right]+ \operatorname{grad} \left[ \frac{\mathrm{e}}{mc} A \cdot 
\frac{m}{\hbar}u\right] \\
& + \frac 12 \frac{\mathrm{e}}{mc} \operatorname{grad} \operatorname{div} A,
\end{align*}
which after simplification is the same as the first equation in (15.2).

For the imaginary part we find
\begin{align*}
&\frac{m}{\hbar} \left( \frac{\partial v}{\partial t}
 + \frac{\mathrm{e}}{mc} \frac{\partial A}{\partial t}
\right) = \\
& \frac{\hbar}{2m} \left[ \frac{m}{\hbar} \operatorname{grad} \operatorname{div} u+ \operatorname{grad} 
\frac{m}{\hbar} u {\cdot} \frac{m}{\hbar} u- \operatorname{grad} \frac{m}{\hbar} \left(v+ 
\frac{\mathrm{e}}{mc} A\right)\!{\cdot} \frac{m}{\hbar} \left(v+ \frac{\mathrm{e}}{mc} A\right)\!\right] \\
& \qquad + \operatorname{grad}\left[ \frac{\mathrm{e}}{mc} A \cdot \frac{m}{\hbar} 
\left(v+ \frac{\mathrm{e}}{mc} A\right)- \frac{\mathrm{e}^2}{2m \hbar c} A^2- \frac{\mathrm{e}}{\hbar} \varphi \right].
\end{align*}
Using (15.9) and simplifying, we obtain
\[
\frac{\partial v}{\partial t}
= \frac{\mathrm{e}}{m}E+ \frac{\hbar}{2m} \operatorname{grad} \operatorname{div} u+ 
\frac 12 \operatorname{grad} u^2- \frac 12 \operatorname{grad} v^2 .
\]
Next we use the easily verified vector identity
\[
\frac 12 \operatorname{grad} v^2=v \times \operatorname{curl} v+v \cdot \nabla v
\]
and the fact that $u$ is a gradient to rewrite this as 
\begin{equation}
\frac{\partial v}{\partial t}
= \frac{\mathrm{e}}{m} E-v\times \operatorname{curl} v+u \cdot \nabla u-v \cdot \nabla v+ 
\frac{\hbar}{2m} \Delta u .
\end{equation}
But $\operatorname{curl} \,(v+ \mathrm{e} A/mc)=0$, since the generalized momentum $mv+ 
\mathrm{e} A/c$ is by assumption a gradient, so that, by (15.8), we can substitute $\mathrm{e} H/mc$ 
for $\operatorname{curl}v$, so that (15.12) is equivalent to the second equation in (15.2) with $F=ma$.

\newpage
%~ \\[0.5cm]
\begin{center}
\textrm{\bf References}
\end{center}

There is a long history of attempts to construct alternative theories to quantum mechanics 
in which classical notions such as particle trajectories continue to have meaning.

\smallskip

\noindent [46]. L. de Broglie, 
``\'Etude critique des bases de l'interpr\'etation actuelle de la m\'ecanique ondulatoire'', 
Gauthiers-Villars, Paris, 1963.

\smallskip

\noindent [47]. D. Bohm, 
\emph{A suggested interpretation of the quantum theory in terms of ``hidden'' variables}, 
Physical Review \textrm{85} (1952), 166--179.

\smallskip

\noindent [48]. D. Bohm and J. P. Vigier, \emph{ Model of the causal interpretation of quantum theory 
in terms of a fluid with irregular fluctuations}, Physical Review \textrm{96} (1954), 208--216.

\smallskip

The theory that we have described in this section is (in a somewhat different form) due to F\'enyes:

\smallskip

\noindent [49]. Imre F\'enyes, 
\emph{ Eine wahrscheinlichkeitstheoretische Begr\"undung und Interpretation der 
Quantenmechanik}, Zeitschrift f\"ur Physik \textrm{132} (1952), 81--106.

\smallskip

\noindent [50]. W. Weizel, 
\emph{ Ableitung der Quantentheorie aus einem klassische kausal determinierten Model}, 
Zeitschrift f\"ur Physik \textrm{134} (1953), 264--285; Part II, \textrm{135} (1953), 270--273; 
Part III, \textrm{136} (1954), 582--604.

\smallskip

\noindent [51]. E. Nelson, \emph{ Derivation of the Schr\"odinger equation from Newtonian mechanics}, 
Physical Review \textrm{150} (1966).

\newpage

\fancyhead[CO]{COMPARISON WITH QUANTUM MECHANICS}
\fancyhead[CE]{CHAPTER \thechapter}
\fancyhead[RO,LE]{\thepage}

\chapter{Comparison with quantum mechanics}

\hspace{6mm}We now have two quite different probabilistic interpretations of 
the Schr\"odinger equation: the quantum 
mechanical interpretation of Born and the stochastic mechanical interpretation of F\'enyes. Which 
interpretation is correct?

It is a triviality that all measurements are reducible to position measurements, since the outcome 
of any experiment can be described in terms of the approximate position of macroscopic objects. 
Let us suppose that we observe the outcome of an experiment by measuring the exact position at a 
given time of all the particles involved in the experiment, including those constituting the apparatus. 
This is a more complete observation than is possible in practice, and if the quantum and stochastic 
theories cannot be distinguished in this way then they cannot be distinguished by an actual experiment. 
However, for such an ideal experiment stochastic and quantum mechanics give the same probability 
density $| \psi |^2$ for the position of the particles at the given time.

The physical interpretation of stochastic mechanics is very different from that of quantum 
mechanics. Consider, to be specific, a hydrogen atom in the ground state. Let us use Coulomb units; 
i.e., we set $m= \mathrm{e}^2 = \hbar =1$, where $m$ is the reduced mass of the electron and 
$\mathrm{e}$ is its charge. The potential is $V=-1/r$, where $r= |x|$ is the distance to the origin, 
and the ground state wave function is
\[
\psi = \frac{1}{ \sqrt{ \pi}} e^{-r} .
\]
In quantum mechanics, $\psi$ is a complete description of the state of the system. According to 
stochastic mechanics, the electron performs a Markov process with
\[
dx(t) =-\frac{x(t)}{|x(t)|} dt + dw(t),
\]
where $w$ is the Wiener process with diffusion coefficient $\frac 12$. (The gradient of $-r$ is 
$-x/r$.) The invariant probability density is $| \psi |^2$. The electron moves in a very complicated 
random trajectory, looking locally like a trajectory of the Wiener process, with a constant tendency 
to go towards the origin, no matter which direction is taken for time. The similarity to ordinary 
diffusion in this case is striking.

How can such a classical picture of atomic processes yield the same predictions as quantum 
mechanics? In quantum mechanics the positions of the electron at different times are non-commuting 
observables and so (by Theorem 14.1) cannot in general be expressed as random variables. Yet we 
have a theory in which the positions are random variables.

To illustrate how conflict with the predictions of quantum mechanics is avoided, let us consider the 
even simpler case of a free particle. Again we set $m = \hbar =1$. The wave function at time 0, 
$\psi _0$, determines the Markov process. To be concrete let us take a case in which the computations 
are easy, by letting $\psi _0$ be a normalization constant times $\exp (- |x|^2/2a)$, where $a>0$. 
Then $\psi _t$ is a normalization constant times
\[
\exp \left(-\frac{|x|^2}{2(a+it)}\right) = \exp \left(-\frac{|x|^2 (a-it)}{a^2+t^2}\right) .
\]
Therefore (by Chapter 15)
\begin{align*}
u & =-\frac{a}{a^2+t^2} x, \\
v & =\phantom- \frac{t}{a^2+t^2} x, \\
b & =\phantom- \frac{t-a}{a^2+t^2} x .
\end{align*}
Thus the particle performs the Gaussian Markov process
\[
dx(t) = \frac{t-a}{a^2+t^2} x dt + dw(t),
\]
where $w$ is the Wiener process with diffusion coefficient $\frac 12$.

Now let $X(t)$ be the quantum mechanical position operator at time $t$ (Heisenberg picture). That is,
\[
X(t) = e^{\frac 12 it \Delta } X_0 e^{- \frac 12 it \Delta },
\]
where $X_0$ is the operator of multiplication by $x$. For each $t$ the probability, according to 
quantum mechanics, that if a measurement of $X(t)$ is made the particle will be found to lie in a 
given region $B$ of $\Bbb{R}^3$ is just the integral over $B$ of $| \psi _t|^2$ (where $\psi _t$ is 
the wave function at time $t$ in the Schr\"odinger picture). But $| \psi _t|^2$ is the probability density of 
$x(t)$ in the above Markov process, so this integral is equal to the probability that $x(t)$ lies in $B$.

We know that the $X(t)$ for varying $t$ cannot simultaneously be represented as random variables. 
In fact, since the particle is free,
\begin{equation}
X \left(\frac{t_1+t_2}{2}\right) = \frac{X(t_1)+X(t_2)}{2}
\end{equation}
for all $t_1$, $t_2$, and the corresponding relation is certainly not valid for the random variables 
$x(t)$. Thus the mathematical structures of the quantum and stochastic theories are incompatible. 
However, there is no contradiction in measurable predictions of the two theories. In fact, if one 
attempted to verify the quantum mechanical relation (16.1) by measuring
\[
X \left(\frac{t_1+t_2}{2}\right)\!, \quad X(t_1), \quad X(t_2),
\]
then, by the uncertainty principle, the act of measurement would produce a deviation from the linear 
relation (16.1) of the same order of magnitude as that which is already present in the stochastic 
theory of the trajectories. Although the operators on the two sides of (16.1) are the same operator, 
it is devoid of operational meaning to say that the position of the particle at time $(t_1+t_2)/2$ is 
the average of its positions at times $t_1$ and $t_2$.

The stochastic theory is conceptually simpler than the quantum theory. For instance, paradoxes 
related to the ``reduction of the wave packet'' (see Chapter 14) are no longer present, since in the stochastic 
theory the wave function is no longer a complete description of the state. In the quantum theory of 
measurement the consciousness of the observer (i.e.,~my consciousness) plays the r\^ole of a deus ex 
machina to introduce randomness, since without it the quantum theory is completely deterministic. 
The stochastic theory is inherently indeterministic.

The stochastic theory raises a number of new mathematical questions concerning Markov processes. 
From a physical point of view, the theory is quite vulnerable. We have ignored a vast area of quantum 
mechanics---questions concerning spin, bosons and fermions, radiation, and relativistic covariance. 
Either the stochastic theory is a curious accident or it will generalize to these other areas, in 
which case it may be useful.

The agreement between the predictions of quantum mechanics and stochastic mechanics holds only 
for a limited class of forces. The Hamiltonians we considered (Chapter 15) involved at most the first 
power of the velocity in the interaction part. Quantum mechanics can treat much more general 
Hamiltonians, for which there is no stochastic theory. On the other hand, the basic equations of the 
stochastic theory (Eq.\ (15.2) with $F=ma$) can still be formulated for forces that are not 
derivable from a potential. In this case we can no longer require that $v$ be a gradient and no longer 
have the Schr\"odinger equation. In fact, quantum mechanics is incapable of describing such forces. If there 
were a fundamental force in nature with a higher order dependence on velocity or not derivable from 
a potential, at most one of the two theories could be physically correct.

Comparing stochastic mechanics (which is classical in its descriptions) and quantum mechanics 
(which is based on the principle of complementarity), one is tempted to say that they are, in the 
sense of Bohr, complementary aspects of the same reality. I prefer the viewpoint that 
Schr\"odinger [30, \S14] expressed in 1926:

``\dots It has even been doubted whether what goes on in the atom could ever be described within 
the scheme of space and time. From the philosophical standpoint, I would consider a conclusive 
decision in this sense as equivalent to a complete surrender. For we cannot really alter our manner of 
thinking in space and time, and what we cannot comprehend within it we cannot understand at all. 
There \emph{are} such things---but I do not believe that atomic structure is one of them.''

\end{document}


