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This is to certify that the

thesis entitled

A Dynamical Theory of Generalized Ornstein-Uhlenbeck Processes

presented by
Timothy Allan Wittig

has been accepted towards fulfillment
of the requirements for

_2h...D.._degree in W

I
M6112—

Major professor

Date August 6, 1981

 

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A DYNAMICAL THEORY OF GENERALIZED ORNSTEIN-UHLENBECK PROCESSES

By
Timothy Allan Nittig

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Statistics and Probability
l98l

ABSTRACT

A Dynamical Theory of' Generalized 0rnstein-—Uhlenbeck Processes

Timothy Allan Nittig

In recent years there has been interest in the generalized
Ornstein-Uhlenbeck «).U.) process as a limit of branching diffusion
processes. We consider the model of the().U. velocity process of
Nelson as a time-change Brownian motion through its covariance. We
generalize this idea through "operator time-change" using one-parameter
operator semi-groups, thereby obtaining a "space-time interactive"
process. We show that in a special case this process coincides with
the generalized «).U.) process of Holley and Stroock who studied it as
a solution of the Martingale Problem. This also gives the generalized
0.lL process of Dawson. We derive a rigorous generalized stochastic
differential equation for this process in the sense of Ito and Kuo.
Using the differential equation and evolution operators we can obtain
a time non-homogeneous generalizedl).U. process as a solution to a
stochastic evolution-type equation. One special case of this involves
non-linear stochastic differential equations in chemical-reaction
problems. We also give a dynamical construction of the Brownian motion
(in the generalized sense) using displacement processes. Finally,
we give appropriate analogs of the infinitesimal generator and Ito

formula of these generalized processes.

Dedicated to the Memory of
My Grandfather
The Reverand Otto H. Nittig
and
To My Parents

"If a thing is difficult to be accomplished by thyself, do not think
that it is impossible for man: but if anything is possible for man
and conformable to his nature, think that this can be attained by

thyself too."
- Marcus Aurelius, "Meditations"

ii

ACKNOWLEDGEMENTS

I wish to thank Professor v.5. Mandrekar, my thesis advisor,
for his guidance during preparation of this dissertation.
I also wish to thank Clara Hanna for her fast and excellent

typing of this manuscript.

TABLE OF CONTENTS

Chapter Page
0 INTRODUCTION........ ........ . .................... 1
I THE RESTRICTED LANGEVIN EQUATION WITH MARKOV
SOLUTION ....................................... 4
1.1 Introduction ................................ 4
1.2 Basic Concepts ..... . ........................ 6
1.3 Generalized Gaussian Processes .............. 9
1.4 Generalized O.U. Processes .................. 14
1.5 Generalized Wiener Processes ................ 18
1.6 A Restricted Langevin Equation... ........... 22
II A DYNAMICAL THEORY OF BROWNIAN MOTION... ......... 32
2.1 Introduction ............................... 32
2.2 Generalized Wiener Set Processes ............ 33
2.3 Generalized Stochastic Integrals ..... . ...... 36
2.4 Stationary Solutions of the Generalized
Langevin Equation .......................... 45
2.5 The Generalized Displacement Process ........ 53
III GENERALIZED EVOLUTION EQUATIONS AND THE
CHEMICAL-REACTION MODEL ....... . ................ 58
3.1 Introduction... ..................... . ..... 58
3.2 Generalized Stochastic Evolution Equations.. 59
3.3 The Chemical- Reaction Equation. ... ....... 63
3.4 Infinitesimal Generators of GGP's .and
.Ito's Formula ........... . ....... ....... ..... 64
APPENDIX A.1 .................. . ......................... .... 74

BIBLIOGRAPHY ........................... ....... .............. 77

iv

CHAPTER 0. INTRODUCTION

The first phenomenological dynamic description of Brownian
motion using the Langevin equation was given by Ornstein and Uhlenbeck
(tom) in 1930 and further work was done by Uhlenbeck and Wang (mm)
in 1945; their results were in agreement with theearlier (1905 and
1906) work of Einstein and Smoluchowski (see [N]). Although Doob
([001]) in 1942 gave the Langevin equation itself a mathematically
rigorous interpretation, Ford, Kac and Mazur ([FKMJ) in 1965 were the
first to give a mechanical model yielding the required behavior of the
Brownian particle. The FKM model consists of a system of coupled
harmonic oscillators in a heat bath with a constant absolute temperature;
however, a stationary description (in accord with the Liouville theorem
of mechanics, see [FKM]) is dependent upon an infinite-particle (i.e.,
infinite-dimensional) system.

In recent years there has been interest in the study of the
Langevin equation in the infinite-dimensional case from the modeling
of physical phenomena described by a system of non-linear (partial)
differential equations on a one-dimensional space by linear operators
on an infinite-dimensional space. The physical phenomena so described
include quantum mechanics ([FKM], [BK]), chemical-reaction problems
([ACK]. [C]), stochastic estimation and control ([CJ), turbulent
transport with diffusion ([Chl), the Fokker-Planck and heat equations
([MiJ), the fluctuation-dissipation theorem ([0]) and scattering theory

([LPJ). In the work of [HS] and [0] the processes involved arose as
weak (renormalization) limits of certain branching phenomena. Also,
the non-homogeneous case studied in [ACK] and [C] for the chemical-
reaction problem was motivated by the work of P. Chow ([Chj) in tur-
bulence theory.

In this thesis we have undertaken to study the theory of the
Langevin equation following the work of Ornstein, Uhlenbeck ([OUJ)
and Nelson (ENJ). We consider the model of the Ornstein-Uhlenbeck
«)Ug) velocity process of Nelson as a time-change Brownian motion through
its covariance. We then generalize this idea through "operator time-
change" using one-parameter operator semiegroups, thereby obtaining a
"space-time” interactive process. Its existence with a C([O,w ), 3')-
version is proved by using a theorem of Prokhorov in [Sc]. This process
is shown to be a generalized process with a ("space-time interactive")
Markov property and can be related to a restricted Langevin equation.
This is demonstrated by an appropriate generalization of Wiener's
stochastic integral. As a consequence of this we prove the uniqueness
of the Martingale Problem in [HS].

The second chapter is devoted to the construction of the so-
called "non-interactive space-time" Brownian motion through a dynamical
model. This work constitutes an exact generalization of the fundemental
work of Ornstein and Uhlenbeck (see also (NJ). The main technical tool
used is an appropriate generalization of the definition of the stochastic
integral. Although the ideas were already given by Ito ([1]), Ito's
suggestion of the work in Kuo (EKJ) depends on the Karhfinen-Loéve

expansion in our context. Fortunately this is available in the recent

work of H. Sato ([SaJ). As a consequence of this construction we can
show that the Holley-Stroock generalized O.U. process ([HS]) is a
projection of a stationary process. ,This fact is crucial in the study
of interacting systems (see e.g. [FKM], [LT 1], [LT 2]). Although
one can relate our work to the so-called LP-structures of Lax and
Phillips ([LPJ) we have not bothered to do so as the arguments are
very similar to those in [LT 2].

Finally, we study solutions of evolution-type equations. Be-
cause the solutions take values in the Suslin space S', the theorem
in [Sc] allows us to construct strong S'-valued solutions. This improves
the work in [Ch], [C] and [ACK]; the Markov property of the solutions
follows immediately. The main tool in this context is again the definition
of the stochastic integral for Gaussian processes and the use of re-
producing kernel Hilbert spaces. Furthermore, we are able to define
the concept of an infinitesimal generator of this S'-va1ued process
with the domain consisting of tame functionals. We also prove an Ito
formula for this context. These results improve the work in [C] and

LACK].

CHAPTER I
THE RESTRICTED LANGEVIN EQUATION WITH MARKOV SOLUTION

51.1. Introduction.

 

If {W(t): t 3 O} is a normalized (W(O) a O) real-valued
(R-valued) Wiener process with drift zero (EW(t) = O for all
t 3 O) and variance parameter 02 (E W(t)W(s) = (t A s)oé, o > O,
for all t, s 3 O) ([F], p. 36) then the stochastic differential

Langevin equation (for a Brownian particle in a free field)
(1.1.1) V(dt) = -8 V(t)dt + W(dt); V(O) a V0, 3 > O

has a unique solution given by the ll-valued Ornstein-Uhlenbeck (O.U.)

velocity diffusion process {V(t): t 3 O, V(O) a V0} where

t
(1.1.2) V(t) = v 'Bt + f e'8(t'u)W(du)
0

0e

In particular, the random variables V(t) are Gaussian with mean

Voe’st and covariance
2 -B|t-s| -B(t+s) 2 “5 -B(t-u) -B(s-u)
(1.1.3) r(t, s) =(o /28)(e -e ) = o f e e du
O

which implies
(1.1.4) ' E(V(t))2==(oz/28)(l—e-28t)

(Further details concerning the classical theory of the Langevin equation

and O.U. processes can be found in [Br], Pp. 347-352 or [N], pp. 53-61;

4

a phenomenological account can be found in [0U] and [WU].)

In this chapter we will motivate and define the concept of
a "generalized" Gaussian process. By a "generalized" process we
essentially mean a process dependent on two parameters (denoted by t
and b), one of which (t) is viewed as a time parameter and the other
(e) as a space parameter. The set of space parameters will be the
Schwartz space of rapidly decreasing functions and, for this chapter,
we restrict the time parameter to III = [0, m'). (We will sometimes
refer to a "generalized" process as an infinite-dimensional space-
time process.) Consistency with the classical theory of stochastic
processes is achieved by requiring that the "generalized" process be
viewed as a functional-valued process (in the usual sense of Kolmogorov)
indexed by t in the Schwartz dual space and, for each o, as a ll-
valued Gaussian process indexed by t.

In order to extend the theory of the Langevin equation to the
notion of a "generalized" process we proceed by first showing that the
Gaussian process corresponding to a particular covariance (similar
to (1.1.3)) is in fact a "generalized" Gaussian process. This is
accomplished by use of the Minlos-Bochner theorem and Prokhorov's con- I
sistency' theorem for Suslin spaces. We then give a functional analog
of the integral form of (1.1.1). By postulating that the solution is
analogous to (1.1.2) we are led to consider functional stochastic
integrals. However, we will defer the general definition of a stochastic
integral until Chapter II, and instead examine the simpler (and more

historical) "integration-by-parts" solution.

§1.2. Basic Concepts.
For the sake of completeness we include the following basic

concepts. Further details can be found in the references cited.

1.2.1. Definition ([M l], p. 19). Let T be an arbitrary set.

 

A Il-valued function C on T x T is called a covariance if

 

(a) C(t, s) = C(s, t) for all t, s e T and

(b) E a a C(t, s) 3 0 for all finite subsets T0 : T and for

t s
t.s€T0

all (at: t6 TD} 5 R.
A function satisfying (a) is called symmetric and if it satisfies (b)

it is called positive-definite.

 

1.2.2. Theorem ([M 1], pp. 7, 8). Let T be a set and C a covariance
on T x T. Then there exists a unique Hilbert space H(C) of functions
on T such that

(a) C(t, -) 6 H(C) for all t 6 T and

(b) for each f E H(C) and t e T, <f, C(t, ')>H(C) = f(t).

The Hilbert space H(C) is called the reproducing kernel Hilbert space

(RKHS) of C.

1.2.3. Theorem ([M 1]. PP. 7-13). Let T be a set. A function C
on T x T is a covariance if and only if there exists some Hilbert

space H and a function f: T + H such that C(t, s) = <f(t), f(s)>H.

1.2.4. Definition ([M l], p. 24). (a) A Il-valued random variable

 

XO defined on a probability space (a, F, P) is called centered

Gaussian with variance 02 (o > 0) if for all Borel sets B c It,

2

c -l’

PoX 1(13) = (21roz) if exp(-1§ L )du
o B 2

O

or, equivalently, if

EEexp(it X0)] = exP(-% tzoz)

(b) We say that X is a Gaussian random variable if for some m e I?

there exists a > 0 such that Po(X-m)'1 = Foxgl.

(We will sometimes
express this by writing X ~ N(m, 02).)

Without loss of generality, we assume that all probability spaces

(a, F, P) are complete; i.e., if B 6 F, P(B) = O and A c B then,

A e F and P(A) = 0.

1.2.5. Definition ([M l], p. 25). Let (a, F, P) be a probability

 

space.

(a) A linear manifold M : L2(n, F, P) is called a Gaussian manifold

I

if x e M implies Po(x-m)“ = Pox; for some m e R, o > o.

(b) An LZ-closed Gaussian manifold is called a Gaussian space.

 

1.2.6. Theorem ([M l], p. 25). Let H be a real Hilbert space.
Then there exists a probability space (9, F, P) and an isomorphism

n mapping H onto a Gaussian subspace of L2(n, F, P).

1.2.7. Definition ([M l], p. 27). A family {X(t): t e T} of

 

ll-valued random variables on a probability space (9, F, P) is called

a Gaussian proceés (GP) if for each finite subset Tb : T and

 

(at: t e To} 9 R , X atX(t) is Gaussian.

tGTb

1.2.8. Corollary. Given a covariance C on T x T there exists a
unique (centered) GP {X(t): t e T} such that C(t, s) = E X(t)X(s).
("Centered"implies E X(t) = O for all t 6 T.)

Proof. Immediate from theorems 1.2.2 and 1.2.6 with «(C(t, -)) =
X(t). Q.E.D.

1.2.9. Remarks. Let T be a set, C a covariance on T x T and
{X(t): t e T} the associated GP given by corollary 1.2.8. Also
let H(C) be the RKHS of C and (a, F, P) the underlying probability
space. It then follows (by theorem 1.2.6) that there exists an isomorphism
mapping H(C) onto the Gaussian subspace L(X) = EEIX(t): t E T}
of [2(9, F, P) where n(C(t, -)) = X(t). Moreover, E|X(t) - X(s)|2
+ O as t + s if and only if "C(t, .) - C(s, ')“H(c) + 0 as t + 5.
Since C is positive-definite, to prove that C is continuous it
suffices ([M l], p. 27) to prove that C(t, t) is continuous.

Let {X(t): t 6 T} be a GP. If T is a linear space and
X(t) is linear in t the GP will be called a linear GP. If

 

tn + t (in the topology of T) implies E|X(tn) - X(t)|2 + 0 we

call the GP continuous. If the index set T is a linear space and

 

C a continuous bilinear covariance on T x T then the associated GP

will be called a Gaussian field ([M 2], p. 24). In this case l.2.l(b)

 

is equivalent to C(t, t) 3 O for all t 6 T. Note that a Gaussian
field is obviously a linear GP since by the remarks above we have

forall t,s€T and a,beR

X(at + bs) «(C(at + bs, -)) = n(aC(t, .) + bC(s, -))

aw(Ct, -)) + bn(C(s, -)) = aX(t) + bX(s).

Let S = 3(Ild) denote the Schwartz space of rapidly decreasing

R -valued functions defined on Rd

d

; i.e., the set of infinitely

differentiable functions cp: R + R such that
“pl; 8 = supd lxo‘Df3 p(x)| < + a. for all a, e
’ x612

where a, B are d-tuples of non-negative integers (multi-indices) and

Xa _ xal xad
l ' d
B B
8 3 l 3 d
D = (_) O O . (—)
3X] 3Xd

Let 3 have the topology generated by the semi-norms “-H0 8 (the
Schwartz-topology). Let S' = S'(1¥j) be the topological dual of
S, i.e., S' is the space of tempered distributions. Let 3' have
the weak *-topology. Under this topology we have that S is dense
in 3' under the natural embedding. We let (o, -) denote the
canonical bilinear form linking S and 3'. Also, let <-, -> and

2 = L2(Ild) inner product and norm. Since 3. 3'

have the Hilbert space rigging S c L2 c 3', <-, -> = (-, .) on

2

n.“ denote the L
SXL and <-, -> extends by continuity to S x S' and this extension.
coincides with (-, o). Further details concerning the spaces 3

and s' can be found in [GV] (pp. 20-26). [11 (pp. 187-191). [Ru]

(pp. 135-207) and [RS] (pp. 135-145).

51.3. Generalized Gaussian Processes.
In order to motivate our discussion of "generalized" Gaussian

processes we consider the following example.

10

1.3.1. Example: A Non-interactive Space-Time Process.

Let T = REX S. We know that r(t, 5) (given by (1.1.3))
is the covariance of an O.U. velocity process (for which we will assume
V(O) s O and o2 = 1) on 12+. Let Q(¢, p). be a continuous bilinear

covariance on 3 x3. Then r(t, s)Q(cp, 1);) is acovariance on

r x T. Let
C[(t. cp). (S. 111)] = r(t, S)Q(q>.p)

(1.3.1) = 0(e. p) EAS e'8(t'“)e'8(s'“)ou

(1.3.2) ' = ZAS Q(e‘8(t'“)o, e‘8(5'”)o)du

The GP associated with. C will be denoted by {Vt(o): (t, o) e T}.
Note that 0(p, o) 3 0. By (1.3.1) we have that for t, s 3 O
tAs
(1.3.3) E Vt(p)Vs(¢) = 0(p. p) 5 e 8(We B‘S'“’du
Comparing (1.3.3) and (1.1.3) we note that for each m E S, {Vt(¢): t 3 0}
is simply a ll-valued O.U. process with variance parameter 0(o, ¢I~

So

(1.3.4) E(v.(cp))2 = (%3>Q(¢’ o)(1-e‘28t)

If we let vm(o) = lim v p). then from (1.3.4) it follows that the

t-e t(
distribution of Vw(¢) is Gaussian with mean zero and variance
Q(¢, ¢)(%B ) this is the analog of the Maxwell distribution on S'
([N], p. 33 and [H], pp. 122-124).

Now fix t 3 o and define ct(o, p) = C[(t, o), (t. ()1.

Obviously Ct is a continuous bilinear covariance on S x 3. Thus,

11

since S is a linear space, {Vt(¢): c e S} is a Gaussian field for
each t 3 0. Furthermore, by the Minlos-Bochner theorem (see [GV].

p. 256 or [H]. pp. 116-122) we obtain that for each t 3 0, V 6 S' P-a.s..

t
We noted above that the covariances (1.3.3) and (1.1.3) are
connected with respect to the fact that the associated O.U. processes
differ only in the variance parameter. We wish to point out another
connection between these covariances. If we consider the difference

2

analog of (1.1.1) (where V(O) 2 o, o = 1) ([Br], pp. 347-349).

(1 3.5) AV(t) = V(t + At) -V(t) = -8V(t)At + W(t + At) - W(t)

as our basic equation, then the covariance of AV is given by the

difference analog of (1.1.3). Similarly, if AVt(o) is given by

(1.3 6) AVt(¢) = Vt+At(¢) -Vt(p) = Vt(-81p)At + 0%(p. ¢)[W(t + atl-N(t)]

(where I denotes the identity operator) then the covariance of
AVt(p) is given by the difference analog of (1.3.3). With this view
in mind we can rewrite (1.3.2) as

tAs

(1 3.7) C[(t. ¢O.(S. p)] = 6 0(Tt-u p. T wldu

S-U

tBI on S.

where Tt denotes the one-dimensional semi-group e-
The construction of Vt(¢) illustrates an easy way to obtain

a "space-time" GP (which can in fact be viewed as a classical O.U.

velocity process). The parameter t 3 0 represents time and the para-

meter e 6 S can be thought of as a vector denoting direction in the

infinite-dimensional space S. (S of course is our set of "space

vectors".) To summarize, starting with the covariance (1.3.2) (or

12

equivalently (1.3.7)) we obtained a unique centered GP {Vt(q0: (t. $06 T}

that satisfies (a) V e 3' P-a.s. for all t 3 O, (b) {Vt(o): e E S}

t
is a Gaussian field and (c) {Vt(¢): t 3 0} is a GP for all p e S
(in fact, it is an O.U. process in each direction o e 3).

Since we want our "generalized" (i.e. space-time) processes
to have properties (a), (b) and (c) above, we are led to the following

definition.

1.3.2. Definition. A family {X(t): t e T} of random variables

defined on a probability space (a, F, P) is called a generalized

 

Gaussian process (GGP) if

(a) X(t) is a functional on some linear space L for all t E T P-a.s.,
(b) {X(t)(n): n 6 L} is a Gaussian field for all t e T and

(c) {X(t)(n): t e T} is a GP for all n 6 L.

We can view a GGP as X(t, n) where X(t, .) is a Gaussian field and
X(-, n) is a GP. Obviously the GP corresponding to the covariance
(1.3.2) is a GGP.

The process corresponding to (1.3.2) is not a very satisfactory
example of a GGP in that the space-time parameters (p and t) are
in a certain sense independent. This can be illustrated in two ways.
First, the distribution of V®(¢) is partially determined by the same
space parameter that appears in.any of the random variables Vt(e), t < a,
through the Q(p, p) term in the variance. Second, by comparing
(1.3.7), (1.3.3) and (1.1.3) we see that the effect of the space para-
meter in the process {Vt(¢): t 3 O} is simply directional. That is,

{Vt(¢): t 3 O} is just a Il-valued O.U. process that diffuses along

13

the direction 3. The Vt(¢) process is such that evolution in time
does not effect the evolution through space. There is no interaction
between the space and time parameters.

We will impose a space-time interaction basically by considering
the infinite-dimensional (operator) analogs of (1.3.6) and (1.3.7).

That is, consider analogous equations of (1.3.6) of the form
L
(l 3.8) AVt(¢) = Vt(A¢)At + Q’(¢. plA N(t)

where A is an operator assumed to generate a semi-group Tt (we

may need to assume additiOnal conditions). Mimicking our earlier comments,
we then hypothesize that the covariance of AVt(o) (equation (1.3.8))

is given by the difference analog of (1.3.7) where now Tt is the
semi-group generated by A. Clearly, equations similar to (1.3.8) will
yield processes having a space-time interaction. (As time goes from_ t

to t + At the space parameter will go from T , i.e.,
th(¢) = ATt(¢)dt-)

Using these comments as justification, we consider generalizations

@ to T

t thit‘p

of (1.3.7) of the form in the next theOrem.

1.3.3. Theorem. Let A: S +-S be a bounded linear operator which

admits an extension A} L2 + L2 that generates a semi-group {Tt: t 3 O}
mapping L2 to L2. If Q(¢, p) is a covariance on S x S then

tAs
(1.3 9) . C[(t. p). (S. p)] = 6 Q(Tt_u¢. Ts_up)du

. . +
15 a covar1ance on T x T, where T = 12 x S.

14

Proof. Since C is obviously symmetric we need only verify l.2.l(b)

Let {(tl’ o1)....,(tn, pn)} : T and {a}....,an} : 12. Then

11

i§j=l aiaj C[(t,. p1).(tj. pj)]

I I” < )
= a,a, Q T _ @o, T _ (p. d“

i,j=l 1 j 0 ti u 1 ti u j

I E a. I (u)a.T (u)Q(T ¢., T p.)du > 0

since 0 is a covariance on S x S. Q.E.D.

An immediate corollary that follows by theorem 1.2.3 is:

1.3.4. Corollary. If B: S *'S then

tAs

(1.3.10) 5 <3 Tt-uP’

B Ts_uw>du
is a covariance on T x T.

By corollary 1.2.8 and theorem 1.3.3 there exists a probability
space (a, F, P) and a unique (centered ) GP {X(t, p): (t, p) e T}
on (n, F, P) such that E X(t, e) = 0 for all (t, e) e T and for
(t, cp,),(s, w) 6 T, E X(t, (pH (5, w)= C[(t, cp), (s, 112)] where C is given
by (1.3.9). From now on we will denote the GP corresponding to covariances

of the form (1.3.9) by' {Xt(cp): (t, (9)6 T}. Each random variable

Xt(e) is Gaussian with mean zero and variance (from (1.3.9))

t
(1.3.11) E(Xt(q>))2 = C[(t. p).(t. pl] = g Q(Tt_ur. Tt_u¢)du

51.4. Generalized O.U. ProceSses.

In this section we will consider in detail covariances of the

15

form (1.3.10), giving sufficient conditons to insure that the associated
GP {Xt(¢); (t, e) e T} is in fact an S'-va1ued process. In particular,

we get (a) Xt(') to be a continuous Gaussian field, (b) Xt e S' P-a.s.

and (c) X.(o) to be a GP. To this end we make the following assumptions

1.4.1. Assumptions.

(a) A: S +-S is a bounded linear operator that admits a self-adjoint

2 2

extension A: L + L that generates a strongly continuous con-

traction semi-group {Tt: t 3 0} mapping L2 to L2.
(b) B: L2 + L2 is a bounded linear operator.
Let
tAs
(1.4.1) C[(t, cp),(s. 4)] = (j) <BTt_ucp, BTS_u(p>du

Of course C is a covariance of T x T (corollary 1.3.4), where
T = lt+ x 3.
Fix t 3 0 and define Ct(o, w) = C[(t, e),(t, 3)]. It is

obvious that Ct is a bilinear covariance on S x S (C is symmetric

t
and, since 3 is linear and Ct(¢, ¢) 3 O for all Q 6 S, Ct is a
covariance). In order to show that Xt(-) is a Gaussian field it
remains to show that Ct is continuous in the Schwartz-topology. By
our remarks of 1.2.9 it suffices to prove that Ct is continuous along
the diagonal of S x 3.

Suppose that cn, o e S (: L2) and up" - p“ + 0 as n + m.

then
ICtOpn. <9") - Ct(cp. cm

t 2 t 2
= 1% HBTt_uan du - é “BTt-ui“ dul

16
2 2
(“BTt_u¢nH - “BTt_upH ldu

<

2
llBTt_u(<pn - p)“ du

IA

Oksr'f‘ O‘Hfi

t “BHZHpn - p12 + 0 as n + w

IA

Now, since the inherited Lz-topology of 3 contains the Schwartz-
topology, if follows that Ct is continuous in the Schwartz-topology.
Thus Xt(-) is a Gaussian field; in fact, the same proof above shows
that Xt(-) is a continuous Gaussian field (see 1.2.9) for each
t30.
By the Minlos-Bochner theorem there exists (in the sense of
Gelfand and Vilenkin [GV]. p. 256; see also [H], pp. 116-122) a “generalized"
Gaussian field X on 3. That is, there exists a probability space

t

(o, F, P) and an S'-va1ued Gaussian random variable X such that

t
(e, Xt) = Xt(¢), for all e e S, P-a.s. and for each t 3 O. In
particular, (3, Xt) = Xt(p) P-a.s. for all t in a countable set
(i.e., the set of measure zero does not depend on t).

We denote the characteristic functional of X ([GV]. PP. 260-262)

t
by Lt’ which is defined by Lt(o) = Etexp(i(¢, Xt))]. For the "generalized"

Gaussian field Xt we then obtain

t
(1.4.2) L (op) = expc-a c (o. cm= expt-t I ((01 _ (PllszJ
t t 0 t u
In order to complete the proof that Xt(p) is a GGP it remains

to show that we can view {th t 3 0} as a stochastic process (in the

usual sense of Kolmogorov) of S'-va1ued random variables.

17

1.4.2. Existence of an s'-valued Process.
The next result is analogous to the Kolmogorov consistency

theorem ([Br], pp. 251-253). If 5% t is the coordinate projection
1,..., n

+ {t ,...,t } . ‘
mapping (S')R‘ onto (3') I n we wish to find a probability

12+ 1 1
p on (S') such that u o 5’ = Po(X ,...,X )- .
t] 9 o o e ,tn t] tn

Note that S' is a Suslin space ([Sc], p.115). Let I denote
the set of finite subsets i of 12" directed by set inclusion.

Define a probability Pi on (s')I‘=os' for each i E I by
1

Pi = Po(Xt, t E i)'], i.e., the characteristic functional of Pi is

given by EXPL'li X C[(t, (Pt). (59 (95)].

t,sei
Let oji: (s')J . (3')1 for i cj (i, j e I) be the canonical
projection mapping. Then Pj o p3} = Pi since for all t e i U I.
Xt(o) = (o, Xt) P-a.s.. Thus {Pj, pji: j, i e I} is a projective

system of probabilities.
For any countable subset I0 : It+ the system {Pj, pji: j, i c 10’
j, i e I} is a projective system. Thus, Xt(o) = (o, Xt) P-a.s. for

I
t 6 I0 implies that there exists a probability p1 on (S') 0 such

0

I .
J = Pj for J c10(j€ I), Where pj: (3') 0 + ($1)J
IS the Projection map.

0

Therefore, by Schwartz's theorem ([Sc]. Pp. 291-2) there exists
.4.
a unique probability p on (S')R such that p o 3;] = Pj. Here
. + . .
SJ is the projection map of (3')]? onto (S')J for j E I.

We have proved

18

1.4.3. Theorem. There exists a probability space (0, F, P) and a
family of S'-va1ued random variables {xtz t 3 O} on (n. F, P) such
that (o, Xt) = Xt(o) for all e e S P-a.s. and for each t 3 0.
In particular, the joint distribution of Xt(o), Xs(d) is Gaussian
and
tAs

E Xt(cp)XS((p) = (f) <BTt_ucp, BTs_u¢>du

Thus, we have that each random variable Xt(o) istaussian with mean

zero and variance (by (1.4.1»
2 t 2 t 2
(l 4.3) E(Xt(p)) = g “BTt_u¢H du = g “BTU?“ du

The last equality of (1.4.3) follows by changing the variable.

The process of theorem 1.4.3 will be called the generalized

 

0.LL (velocity) process. This terminology will be justified when we

 

discuss the Langevin equation.

51.5. Generalized Wiener Processes.

 

In order to study the analog of the Langevin equation we need
the concept of a Wiener process in our context.

Let Q be acontinuous bilinear covariance on S x S (i.e.,
Q(o, e) 3 O and 0(o, w) = Q(¢, o)). Since no confusion should result,
we will denote Q(¢, p) by 0(o). Define Cw: T x T +-Il (where
T = It‘ x S) by

(1.5.1) cw[(t. p). (s. ()3 = (t A s)Q(p. v)

Obviously C is a covariance on T x T. Thus, by corollary 1.2.8

N
there exists a GP {(o, W(t)): (t, o) 6 T} defined on a probability

19
space (a, F, P) such that E(p, W(t)) = O and

(1-5-2) E(¢. W(t))(w. W(SI) = (t A S)Q(¢. y)

Since this process will play an essential role in our analysis we will

enumerate its properties in the following theorem.

1.5.1. Theorem. (a) For each o e s, {(p, W(t)): t 3 O} is a nor-
malized Il-valued Wiener process with drift zero and variance parameter
0(4)).

(b) For each t 3 O. {(9. W(t)): p 6 S} is a continuous Gaussian field.
(c) {W(t): t 3 O} is a GGP that can be realized in C([O, m), S').

3399:, (a) Immediate by (1.5.2) since E(¢,W(t))(p, W(s)) = (t A s)Q(p).
(b) Immediate since 0 is a continuous bilinear covariance on S x S.
(c) By (a), (b) the family {W(t): t 3 O} is a GGP and furthermore,

by the Minlos-Bochner theorem, W(t) e S' P-a.s. for each t 3 0. Also
an argument similar to that for theorem 1.4.3 proves that there exists
an.SV-valued stochastic process Wt such that the distribution of
((p, N(t])),...,(¢, W(tn))) is Gaussian and is the same as that of

(Wt (cp),...,Wt (p)) for all t1,...,t > 0.. It remains to show
1 n n '

that W(t) has continuous sample paths. For p e S we have
4 _ 2 2
El(<p. W(t)) - (Cp. W(SHI - 3[El(<P. W(t)) - (<9. 14(5))! J

ewe;wor+so.mnF-2uewono.none

utmw+sow)-utwno(wf

30efi<dhfi

20

Thus, by [V], p.28, it follows that (p, W(t)) has continuous sample

paths (in the usual sense) for each p e S. For any linear functional

y on S to be in 3' it must be continuous. But this is equivalent
([RS], pp. 129-134) to the existence of a semi-norm “'“o,8 with

|y(q1)| 5c llcpllChg for _c > O for all cp E S , if we define (Mia’s

by 10113,), = supilhp. M: p 6 8. 114111.M3 5 1} then E(llw(t) - N(S)II;,B)
f (canstant) lt-sl2 as (Q(<p))2 f (constant) llcpllza’s for all multi-

4

indices a. B by the continuity of Q on S. If 33,8 denotes S'
with the topology generated by Monk’s then u*{C([0, on), 36,8 )1 = l,
where u* is the outer measure of the probability u corresponding to
W(t) guaranteed by 1.4.2. But this is true for all multi-indices a. 8
and since f E C([O,w), 3') implies there exists an a, B such that
f e 0([0, e), 3'”) we obtain p*{0(c0.-), s')} = 1. By [Br] (pp.
255-257) this shows that W(-) E C([O, m), S') for P-almost all
m E n, i.e., W(t) has continuous sample paths. Q.E.D.
ll word of explanation is necessary here. We have used the notation
"(p, W(t))“ for the process in order to emphasize that W(-) E C([O, a),
S') P-a.s.. From now on we choose the version (p, W(t))(f) = (p, f(t))

for f e C([0, a), 3'). As an immediate consequence of l.5.l(a) we

obtain:

1.5.2. Corollary. If F = o{(p, f(s)): f e C([O, m), S'), O 5 s 5 t, p e S}

t
and F = o( U Ft) then (o, W(t)) and (p, W(t))2 - t 0(p) are
t>O .
F -martinga1és.

t

1.5.3. Corollary. Let Ft, F be as in corollary 1.5.2 and let
{W(t): t 3 0} be an S'-valued Gaussian process. Then the following

are equivalent:

(a)

(b)

(C)

(d)

21

)2 - t0(p) are F -martingales for each

(r. W(t)). (e. W(t)
9’6 3.

t

For each p e S, (p, W(t)) is a (centered) Wiener process
with variance parameter Q(p).
t
FWEH fed<khfupwhn)e)fww.wpn« man
0

F -martinga1e for each p E S.

t
For each p E S, exp[i(p, W(t))+ % tQ(p)] is an Ft-martingale.

Proof. Exactly as in ll-valued case; see [SV], pp. 82-84. Q.E.D.

1.5.4. Remark. It follows from corollary 1.5.3 that the Gaussian

process {W(t): t 3 0} will have covariance E(p, W(t))(w, W(s)) =

(t A s)O(p, 3). To see this note that if 0 5 s < t < a then by

l.5.3(a)

Howonh.mu)=uowoxhwon
and

upwon2=tmw
But

Ema. 11(5)) - (p._t~1(s))12

Ew4.mur

50(4-4)

s[Q(¢) + 0(4) - 20(p. 4)]

22

and
EH41. W(s)) - (1.14st
=uewon2+unwou2-xw.wuwnwon
= sow) + sow) - 220p. W(s))(w. 11(5))
Thus

E(CP. W(t))(v. 14(5)) = 15(4). "(5))(10. ”(SD = $009.11))

Then, by l.5.l(c) we know that W(t) can be realized in C([O, m), 3').
Following Ito ([I], p. 191) we call the process {W(t): t 3 0}
corresponding to the covariance (1.5.1) a generalized Wiener process.

If we let 8: L2 + L2

be a bounded linear operator and define Q(p, p)
= <Bp, 83> then (p, W(t))~ N(O, t“B¢“2). In this case we say that
W(t) is a generalized Wiener process with drift characteristic zero
(i.e., the zero operator) and variance characteristic 8. If we let
0(p, p) = <¢. p> we call W(t) the standard generalized Wiener process.

(W(t) has drift characteristic zero and variance characteristic I -

the identity operator).

§l.6. A Restricted Langevin Equation.

 

In section 1.3 we suggested that the appropriate form for the
"generalized" Langevin equation (written as a difference equation)
would be given by (1.3.8). In this section we will consider a restricted
version of (1.3.8) and examine some of the properties of its solution.
We will begin by reviewing some properties of semi-groups and vector-

valued integration.

23

1.6.1. Definition ([HP], p.58). Let [a, b] c It and let L denote

 

a Banach space. A function f: [3, b] +‘L is strongly continuous

 

at t 6 La. b] if lim Hf(t)-f(t0)llL = 0.

Let "n = {a = t0 <---< tn = b} denote a partition of [a, b], Innl =

)

0

n
'5tj we call 2 f(sJ.)(tJ. - t

ma t. - t. . F t. s
x I J .1-1l °r l 5 J j:]

J- 3-1
a Riemann-Stieltjes sum. If the limit of the Riemann-Stieltjes sum
exists in the norm-topology of L as n 3.3 and Innl + O we call
the limit the integral I f(s)ds relative to the norm-topology
([HP], pp. 62-63). The :ext lemma concerns the existence of the

integral in the norm-topology of L.

1.6.2. Lgmmg_ ([HP], p. 63). Let f: [3, b] + L be strongly continuous,
then I f(u)du exists in the norm-topology of L. 'Furthermore, if
S is 2 closed linear operator mapping L to L and f(u) e D(S)
for u e [a, b] then S T f(u)du = I S f(u)du.
a a

t
1.6.3. Lemma ([E ], p. 16). Under assumption l.4.l(a), f Tu p du 6 D(A)
O

= S for all p 6 S and
t t t
Ttp-o = A f 1 odd = f A 1 odu = f 1 Aodu
0 u 0 “ 0 “

Assume that l.4.l(a), (b) both hold. Let {xtz t 3 0} be
the GGP with covariance (1.4.1) and let {W(t): t 3 0} be a generalized
Wiener process with covariance (1.5.1) where 0(3, p) = <Bp, 84>.

Let

t
(1.5.1) vt(o) = Xt(<p) - é Xs(Acp)ds

24

for all p 6 S. (For the sake of simplicity we will assume throughout

this section that all processes are n0rmalized to be zero at t = 0.)

1.6.4. Lemma. EYt(p)YS(w) = (t A s) <Bp, Bw> for all 9. w E S and
t, S 3 0.

Proof. Without loss of generality assume 0 5 s < t < m. Then

EYt(¢)Ys(w)
(1.6.2.3) = EXt(p)XS(w)
s
(l.6.2.b) - EXt(p) é Xu(Aw)du
t
(l.6.2.c) - sxs(p) f xu(Ao)du
O
t
(1.6.2.d) + E f 7 Xu(A¢)Xv(Aw)dudu
O 0
By (1.4.1),
(1.6.3.3) (1.6.2.3) = I (BTt-uv’ BTS_un>du

0

By Fubini's theorem ([002]. PP. 430-431),

-(1.6.2.b) =

m
x
4"
A
'6
V
x

u(Aw)du

<BTt_vp, BTu_vA0>dvdu by (1.4.1)

(‘ham O‘e:

<8Tt_vp, BTU_vAw>dudv by Fubini's theorem

<BTt_v@. B(Ts_vw-4)>dv

0%“ OR.“ 0%“ 0%”

by lemma 1.6.2 and lemma 1.6.3 and Fubini's theorem. Thus,

(1.6.3.b)

-(1.6.2.b) =

25

S
g <BTt_vp, BTS_Vw>dv

S
- 6 (BTt-v?’ Bw>dv

Similarly, for (l.6.2.c) we obtain

(1.6.3.c)

Finally,

(1.6.3.d)

-(1.6.2.c) =

0“”, O

(1,6.2.d) =

t
g EXU(A¢)XS(w)du

t UAS

Kb
%

<BTu_vAp, BTS_vp>dvdu

<BTu3vAp, BTs_vw>dvdu

0%: O

(BTU-OA?’ BTS_vw>dvdu

+
“Ru-l"
0%“

0%”

C‘s”

<BTu_vA¢, BTS_VW>dUdV

<BTu_vA¢, BTS-vw>dudv

+
0%”
“Rafi

5
£ <B(TS_Vp-p), BTS_Vp>dv

S
+ g <B(Tt_vQ-Ts_v¢)o BTs_vw>dv
S
g (BTt-vw’ BTS_Vw>dv

S .
- g <Bm, BTS_Vw>dv

EXu(Ap)XV(Aw)dvdu

>
C

<BTu_0Ao, BTv_OAw>dodvdu

O‘fim O‘fiffO‘St‘f
0%: 0%” 0‘5”
Oksc 0‘s:

<BTU_OA¢, BTv_OAw>dodvdu

(l.6.3.e) + <31 _vo, BTV_GAw>dodvdu

(1.6.3.f) + <BTU_OA¢, BTv_ko>dodvdu
But,

<BTU_OAp, BTv_OAw>dvdodu

0“:

<BTu_oAp, B(Tu_Ow-w)>dodu

<BTU_OAp, BTu_Ow>dodu

ll
0%” 0%” OR.”
0‘s: 0“: CH5:

0%”
0‘3:

<31u_vo, Bp>dodu

<31u_vo, BTu_op>dodu

II
0“”
CD‘S:

<BTu_oAp, Bw>dudo

I
O‘Hm
Qk—sm

ll
Ok-Jn
(DH-a:

<BTu_OA¢, BTu_Cp>dodu

<B(Ts_o¢-¢). Bw>do

O‘fim

(BTU-OAQ’ BTu_Cw>d0dU

ll
0‘.“
O\C

s
(1.6.3.9) - f <BT p. Bw>do + S<Bp. Bp>
0 5-0
And

(l.6.3.e) =

c%m

(BTu-ko’ BTV_cAp>dvdodu

<01u_vo, B(TS_Op-Tu_cw)>dodu

0%” 0‘5“
0“: 0%:

(1.6.3.b)

And

(1.6.3.1)

(1.6.3.f) =

0&1” (DR—.10 9%”, ORV, O‘HU’ O‘HU’ Mk‘rf mkxfl

27

s s
g g <BTU_OA¢, BTS_Op>dudo
s u
- 6 6 <BTu_oA¢. BTU_Ow>dodu
s
g <B(Ts_o¢-p), BTS_O¢>do

S u
- g 6 <31u_vo, BTu_ow>dodu

s
f <BTS_Op, BTS_O¢>do
s
- g <B¢, BTs_cw>do
s u
- 6 6 <31u_vo, BTu_Ow>dodu

Qk‘hm

(BTu-OAQ’ BTv_OAw>dvdodu

_0Ap, B(TS_Op-w)>dodu

in-u+c3~501cr~sm
A
W
_g
C

<BTu_0Ap, B(Ts_Op-p)>dudo

A
a:
A
.4
('1'
I
G

5-0?)’ 3(Ts_c¢‘¢)>d0
(BTt-o?’ BTS_Ow>do
(BTt-o?’ Bw>do

<BTS_OQ, BTS_Ow>do

<BTS_Gp, Bp>do

28

Now combining (1.6.2.3) through (1.6.2.d) with (1.6.3.3) through
(1.6.3.1) we obtain EYt(3)YS(3) = s<83, 83> and hence the lemma
follows. Q.E.D.
Noting that lemma 1.6.4 implies that {Yt: t 3 O} is an
S'-valued generalized Wiener process (see section 1.5) then corollary

1.5.3 holds for Yt(3) and by theorem 1.5.1 we immediately obtain

1.6.5. Corollary. The GGP {th t 3 O} with covariance (1.4.1)
can be realized in C([O,m), 3').

Let us denote by P the measure induced by {th t 3 O} on
C([O, m), S'). Then by lemma 1.6.4 and remark 1.5.4 we get

1.6.6. Corollary. There exists a unique measure P on C([0, m), S')
such that Yt(3) satisfies 1.5.3(c).

1.6.7. 333355, (a) Let {xtz t 3 0} be the sample-continuous GGP
with covariance (1.4.1) on the probability space (a, F, P). Then by
lemma 1.6.4 there exists a generalized Wiener process {W(t): t 3 O}
with covariance (t A s) <83, 83> on (n, F, P) such that (p, W(t)) =
Yt(3) P-a.s.. Furthermore, the solution (3, W(t)) = Yt(3) is unique _
by corollary 1.6.6. (This includes theorems (1.4) and (1.23) in the
work of Holley and Stroock in [HS]. PP. 741-751.)

(b) When we speak of unique solutions we of course mean uniqueness

in terms of probability law (or distribution). Although this may seem
insufficient we should recall that in the physical theory of Brownian
motion distribution is the most general thing we can predict (see

LOU]. CWU].[FKM]).

29

1.6.8. Theorem. If

t
(l 6.4) Xt(¢) = (p. W(t)) + £ (Tt_uAp. ”(Ulldu
then Xt(3) solves
t
(l 6.5) (p. W(t)) = Yt(¢) = Xt(¢) - 6 Xu(A¢)du

and E Xt(3)Xs(3) is given by (1.4.1).

Proof. The second assertion follows by lemma 1.6.4 and remark 1.6.7.

To prove the first assertion consider

t
I Xs(Ap)ds
0

r.-

5 2

= I [(A3, W(s)) + f (Ts-uA 3. W(u))dUst
0

(A3, W(s))ds + (ATS_UA3, W(u))duds

(A3, W(s))ds + (A Ts_uA3, W(u))dsdu

CSHO‘em

(Ar. N(S))ds + (T

0%5'? 0%5'? O‘ad'

t_uA3 - A3, W(u))du

(T A4. W(U))du

t-u

>< 0‘54"? ORCPORH‘F Oksfio

t(<p) -- (e. um) 0.5.0.

If we let A': S' +1S' be the adjoint of A defined by

(p. A'y) = (A3, y) for 3 e S, y e S' then we can rewrite (1.6.5) as
' t
(l 6.6) Xt(¢) = 6 A'Xu(¢)du + (p. W(t))

We will formally express (1.6.6) as

30

t
(1.6.7) Xt = 6 A Xudu + W(t)
We can rewrite (1.6.7) in a differential form as
(1.6.8)- dXt = A'Xtdt + W(dt), XO 5 0

We will call (1.6.8) the restricted generalized stochastic differential

Langevin equation (simply, the restricted Langgvin equation). Any

 

solution of (1.6.8) (such as (1.6.4)) will be called a generalized O.U.

(velocity) process (restricted version).

 

1.6.9. Remarks. (3) By our comments of 1.6.7, the process Xt(3)
defined by (1.6.4) is the unique solution (in the sense of distribution)
of the restricted Langevin equation (1.6.5).

(b) If FS is as in corollary 1.5.2 and 0 g s < t < m then
EEXt(p)lFSJ = XS(Tt,sp)

showing that {th t 3 O} is indeed a Markov process.
(c) If X0 2 y 6 3' P-3.s. then the solution of (1.6.8) is easily
seen to be
't

(1.6.9) Xt(p) = (Ttp. y) + (p. W(t)) + g (Tt_ue. W(U))du

In order to gain some insight into the origin of (1.6.4) as
a solution to (1.6.5) let us reconsider for the moment (1.1.1) with
V(O) s 0. As we noted, the solution to (1.1.1) is given by

t
V(t) = 6 e'5(t'”) W(du)

31
Integrating by parts, we can rewrite V(t) as
t -e(t-U)
V(t) = W(t) + f (-8)e "(Uldu
0

It was this form of the solution that was exploited in theorem 1.6.8.

By the covariance (1.1.3) we see that V(t) is a non-stationary
solution which is terms of a physical interpretation is not satisfactory
(see [FKM], pp. 504-510 and [LTl]. [LT2]). We can, however, obtain
a solution to (1.1.1) that is stationary by considering the time-

parameter set to be R . Let {W(t): t E R} be a process such that
E W(t)W(s) = 5(ltl + Isl - |t-s|)
Then the stochastic integral
I e'B(t'")w(du)

is well-defined. The covariance of this integral is easily computed
as %B(e'8|t'sl) which implies that the integral is stationary. Also,
it is easily seen that this integral is indeed a solution of (1.1.1).
So, in order to consider stationary solutions of the generalized
Langevin equation it is necessary to define "generalized" stochastic
integrals of the form

t

[3(Tt-u?’ W(dU))

It is this concept - a "generalized" stochastic integral - that we

will disucss in the next Chapter.

CHAPTER II
A DYNAMICAL THEORY OF BROWNIAN MOTION

52.1. Introduction.

 

As stated in [FKM] (p. 505), "Brownian motion is a thoroughly
studied experimental phenomenon with a statisfactory phenomenological
theory", and as such, in any mathematical investigation of the subject
we should expect and look for results that are in accord with empirical
evidence. For example, in [FKM] the dynamics of a system of coupled
oscillators for a Brownian particle in a heat bath with a constant
absolute temperature are studied and it is observed that the coordinate
and momenta stochastic variables are stationary Gaussian processes
in accord with the Liouville theorem of mechanics ([FKM]. pp. 504-506)
(i.e., the particle suffers no memory effects). Because of these con-
siderations, the present investigation will examine those conditions
necessary to obtain stationary descriptions of the dynamics of 3
Brownian particle; i.e., we will seek stationary solutions of the
Langevin equation.

In [FKM], [LTl] and [LT2] stationary solutions are obtained
for the (appropriately interpreted) Langevin equation essentially by
expanding the space on which the processes are defined. Our method
is quite similar to that found in both [LTl] and [LT2]. That is, we

will consider S'-v31ued processes on 12 and express the solution as

32

33

t
a stochastic integral (written formally) of the form f F(t, u)W(du).

In this chapter we discuss "generalized" stochastic integrals

of the f0rm

(2.1.1) i (F(t, ulcp. N(dU))

where F(t, u) is a bounded linear operator mapping 3 to S for
each t, u in some Borel subset A of 11. (In addition we will
restrict ourselves to considering only those F(t, u) that are deter-
ministic as opposed to random processes).

Before defining (2.1.1) it is necessary to extend, in a consistent
manner, the definition of the generalized Wiener process to 11 (section
2.2). The manner of generalization is similar to that used in [LTl]

(pp. 99-102) and [L12], (p.50). Then, motivated by Ito's definition
([I], p. 193), we construct the stochastic integral in a manner that

is similar to the usual ll-valued construction (see e.g. [F], pp. 55-72)
and mimicks the construction of Kuo's stochastic integral on abstract
Wiener spaces ([K], pp. 188-193).

Once the "generalized" stochastic integral is defined, we will
show that the theory of the Langevin equation can quite easily be
extended to include stationary solutions. Further, this extended theory
is consistent in that it includes the theory of the restricted Langevin

equation (and its solution) as discussed in section 1.6.

§2.2. Generalized Wiener Set Processes.

 

In this section we will extend the notion of the generalized
Wiener process discussed in section 1.5. To motivate our discussion

note that (t A s)Q(3, 3) can be rewritten as X(EO, t] O [0, s])Q(3, 3).

34

where A denotes Lebesgue measure on It, and the corresponding process
can be rewritten as (3, W([O, t])). Thus, we are led to try and define
processes of the form (3, W(A)) corresponding to covariances
X(A n 8)Q(3, 3), where A, 8 are Borel subsets of' 11. (Similar reasoning
is used in both [LTl] and [LT2].)

Let Q be a continuous bilinear covariance on S x 3. Let
(M, M, v) be an arbitrary measure space, where v is a o-finite
measure. Let M0 = {A e M: 0(A) < + a} then 0(A n 8) is a covariance

on M0 x Mo. Def1ne CV on T x T where T = M0 x S by

(2.2.1) CVE(A. <9). (B. 41)] = v(A 0 B)Q(<p. 1))

Then Cv is a covariance on T x T. Let {(3, W(A)): (A. ?) E T}
denote the unique centered GP associated with Cv defined on the
probability space (9, F, P) (see corollary 1.2.8). Then

(3, W(A)) ~ N(O, v(A)Q(3)) by (2.2.1). Denote by H(Cv) the RKHS
of Cv and let n be the isomorphism mapping H(Cv) onto

s‘p'ihp, W(A)): (A, 3) e 1} :- L2((2, F, P). The properties of the

GP {(3, W(A)): (A, 3) 6 T} are enumerated in the following theorem.

2.2.1 Theorem. With the above notation

(a) (-, W(A)) e S' for each A 6 M0.

(b) If A. 6 M0 are mutually disjoint, A = U A
J .1'

j then for each 3 6 3,

(<9. 1101)) ‘1 (o. mp).
J

(c) If A, B 6 M0 and 8 c A then for all 3, 3 E S, E(3, W(A))(3, W(8))
= v(B)Q(¢. y).

(d) If v(A O B) = O for A, B 6 M
E(¢. W(A))(w. “(31) = 0-

0 then for all p. w E S,

35

Proof. (c), (d) are immediate from (2.2.1).
(3) Noting that (p. W(A)) = n[v(A n -)Q(3, -)] it follows that
(-, W(A)) is linear on S. Also, since Qn'T 3 (in the Schwartz-

topology of S) implies Q(¢n. v") +'Q(3. 3) it follows that
(-, W(A)) e S' P-a.s.. (Simply, we say W(A) e S' for each

A e M This is essentially the proof of the Minlos-Bochner

0.
theorem for this case.)
(b) (4. 11(4)) = (o. W(u A31)
J
n£v(u A5 0 -)Q(p. .)3
J
WEZ ”(A3 H °)Q(¢. °)J

"LV(AJ {1 ')Q(CP, ')1

Z
3
=2 (3 N(Aj) 0.5.0.
3
Following [Ml] (p. 28) we call the GP {(3. W(A)): (A, p) e 1}

corresponding to covariance (2.2.1) a generalized Wiener set process.

2.2.2. 539355. In order to see that a generalized Wiener set process
is a consistent extension of a generalized Wiener process (section 1.5)
consider the o-finite measure space (12, 8, A) where 8 denotes the»
Borel subsets of ll and A is the Lebesgue measure on 12. Define
80 = {A e B: X(A) < + m} and

(2.2.2) . . C[(A. p). (B. 3)] = 4(A O B)Q(e. w)

on T x T (T = 80 X 3). Let {(3, W(A)): (A. p) 6 T} denote the

GP associated with (2.2.2). Now, if we let W(t) denote W([O, t])

then the process {W(t): t 3 0} has covariance (t A s)Q(3, 3) and

36

hence is a generalized Wiener process. For this reason (since no
confusion should result) we will often refer to the generalized Wiener
set process corresponding to (2.2.2) as simply a generalized Wiener

process.

52.3. Generalized Stochastic Integrals.

 

In this section we define and examine the properties of a
"generalized“ stochastic integral. By this we mean simply an integral
of an operator-valued (deterministic) function with respect to a

generalized Wiener (set) process, formally f F(t, u)W(du). As in the
A

usual construction, we begin by defining the integral for "simple"
functions and then show that the definition can be extended to a large
class of functions which contains the "simple" functions as a dense
subset.

Let Q be 3 continuous bilinear covariance on S x S. 80 the
Borel subsets of I? with finite Lebesgue measure and {(3. W(A)):
(A, 3) 6 T} the generalized Wiener (set) process corresponding to
covariance (2.2.2) defined on the probability space (a. F, P). Let
B(S) (respectively, B(S')) denote the space of bounded linear operators

mapping 3 to 3 (respectively 3' to 3'). Let

2

L2(Q; R) = {F(t. u): R -> B(S) such that {2 Q(F(t, u)3)du < + ..

forall 116 12,363}

2(03 R) if Q(F(t, -)q.>) E L](du) for each

2(

That is F(t, u) 6 L

t E R and (9 E S. The notation L Q; R) is meant to be suggestive

2

of the usual L notation; Q(-) is a quadratic functional, which we

37

can view as being analogous to “.“2 (Qk(-) is analogous to “-N)
and L2(Q; 12) is that class of (operator-valued) functions whose
“norm" is square-integrable. In fact, it is easy to see that L2(Q; I?)

C H(C), the RKHS of the covariance (2.2.2).

2.3.1. Definition. We say that F E L2(Q; 11) is simple if there

 

exist mutually disjoint sets A1,...,An 6 80 such that F(t, u) = F(t, j)

n
on A3. and F(t, u)= X F(t,j)IA for all t6 R.
i=1 J

If F e L2(Q; R) is simple then the (R-valued) random variable

n
(2.3.1) I (F(t. Jhp. N(AJ-H
.1'=1

exists for each 3 E S. We call (2.3.1) the generalized stochastic

 

integral of F with reSpect to W evaluated at 3 and denote it by

(2-3-2) f(F(t. 01¢. N(dU)) = f(¢. F'(t. U)N(dU))

where F'(t, u) 6 8(3') is the adjoint of the operator F(t, u).
Usually we will write(2.3.2) formally (suppressing the vector of

evaluation 3) as

(2.3.3) [F'(t, u)W(du)

For some Borel subset A c R we define

(2 3.4) {(F(t. u)3. W(dU)) = [IA(U)(F(t. U)¢. W(dU)) = ((IA(U)F(t. Ulp. N(dU))

The random variable (2.3.1) (or, equivalently, (2.3.2)) is
obviously Gaussian with mean zero (it is the sum of independent mean

zero Gaussian random variables) and variance

38

E[f(F(t, u)o. w(du)12

= ,E]E(F(t. J)¢. W(Aj))2 by theorem 2.2.1(d)
J:
n

= I v(AJ-)Q(F(t.;i)q>) by (2.2.2)
i=1

(2.3.5) (0(f(t. u)o)du < + -

Likewise, one can compute the covariance as

EI(F(t. Ulv. W(dU)) f (F(S. u)w. N(dU))

.2. 5(F(t. i)p. N(Ai))(F(s. 1)3. W(Aj))
1.3

z 0(f(t. i)e. f(s. 31¢IV(AJ)
J

(2.3.6) = I Q(F(t. U)<P. F(S. Ulw)du

It follows by 2.2.1(a) that for simple F e L2(Q; R), f(F(t, u)-,
W(du)) e S' P-a.s..

Next we will define (2.3.3) when F is an arbitrary element
of L2(Q; 11). This is accomplished by use of Sato's Fourier expansion
theorem ([53], pp. 1-13) (which is an extension of the Karhfinen-Loéve
expansion ([Ml], p. 27)).

Note first that S' (and S) is a locally convex Hausdorf space.
By the Minlos-Bochner theorem there exists a (centered) Gaussian

(Radon) measure 7 on 3' corresponding to 0 such that

o = 3' (e. y)y(dy) and 0(3) = f (o. y)2y<dy)

39

and 3 +'(¢. .) is Gaussian (i.e., (3, -) ~ N(O, Q(3))). Thus, by

Sato's theorem th e exists {3.} c S' and {n-} G S such that
1 if j = k 3 J
(0-. .) is an independent sequence of

(”'9 E ) = 5' = s
3 k 3k 0 if j r k 3

Gaussian random variables with mean zero and variance one and

(2.3 7) y = 2 (nj. 3'15j i-a.e.(y) (y e 3')
j=l

1.8.,

(2°308) (Q: ') = Z (”is ') ($5 5') Y'a°e°
i=1 3

Define Pa 6 B(S') for n 3 l by
n

(2 3.9) P; y = jg] (nj. Y)€j. y e s'

The next few lemmas investigate the interesting properties of the

operators Pé.

2.3.2. 39333. dim (Pa 3') = n.
3329:, Obvious. O.E.D.
2.3.3. 33333, (Pa)2 = P; .
£3991, Let y 6 S', then

n
P5 (P; y) = P5 (jgI (nj. y)£j)

n n n
= ., P‘ . = ., , .
jg] (nJ y) “:3 .21 (nJ y) k; (nk EJ)Ek
n
= jg] (”j’ Y)£j = Pn y s1nce (nk, gj) = ij Q.E.D.

40

Thus, {Pa: n 3 1} is a family of finite-dimensional operators in

I I2_I II I I
B(S ) such that (Pn) - Pn and PnS : Pn+l S for all n 3 1.

2.3.4. Lemma. If PH 6 8(3) is given by

n

P ¢ = Z (we 5')”

. for 3 E S
n j=1 J J

then P; is the adjoint of the operator Pn’

Proof. For 3 6 S, y e S' we have

n

(Pn ‘P. y) = jg] (3. Ej)(nJ-. y) = (3. P; y) Q-E.D.

8y lemma 2.3.4 it follows that {Pnz n 3 1} is 3 family of finite-
dimensional operators in B(S) such that Pn S : Pn+1 S for all
n 3 1. Furthermore, by Sato's theorem we obtain for all 3 6 3
(<2. 3') = lim (<2. Pr; y) = lim (Pn <9. 1!) rate-(3') (y 6 3')

n+o n+~

Now let F 6 L2(Q; I!) and let Fn(t, u) = PnF(t, u). By
Sato's theorem it follows that for all 3 6 S

(Fn(t. u) 3. -) + (F(t, u) 3, .) y-a.e. for all t, u 6 11

Therefore ([Ml], pp. 24, 25), since (3, -) ~ N(O, 0(3)) for all
3 6 3.

2+0 forall t,u€R

 

EYI(Fn(t9 u)Wa ') ‘ (F(t, U)¢, ‘)

and

EYI(Fn(t, U)¢, ’) ' (F(t, u)¢a ')l2

41

2

EY|(Fn(t9 U)? ' F(t, u)¢a °)|

Q(Fn(t. U)¢ - F(t. U)3)

So, Q(Fn(t, u)3 F(t, u)3) + 0 for all t. u e 11

2<

2.3.5. Lemma. Fn 6 L Q; 11).

Egggf. Obviously Fn(t, u) 6 8(3) for all t, u e 11. It suffices to
show that for each t, u 6 ll and 3 e S, Q(F(t, u)3) 3 Q(Fn(t, u)3).
By (2.3.8), (F(t, u)3, -) = .E] ("j’ -)(F(t, u)3, gj) y-a.e. and so
0(f(t. u)o) = 03:](f(t. u)o.JEj)nj). Thus

Q(F(t. U)¢) - Q(Fn(t. U)3) = Q(F(t. U)3) - Q(Pn(F(t. U)¢))

- ° 2 ”
‘ jg] (F(t, u)¢93€j) Q(nj) + i3j§13i¢j (F(t, U)¢9 €1)(F (to U)¢a Ej)Q(Oia nj)
" 2
' jg] (F(t, U)¢, Ej) Q(Oj)
n
- i,j=I,ifj (F(t. U)3. tj)(F(t. U)3. ti)Q(ni. nj)
= I (F(t, U)¢. a )20(n-) + I (F(t. U)p. ai)(F(t. Ulp. t.)Q(ni. n.)
J=n+1 J J i.j=n+l.ifi J J
. Q(j=§+](F(t. u)3. Ejlnj. j=g+](F(t, u)3, gj)nj) 3 0 0.5.0.

Therefore, the Lebesgue dominated convergence theorem implies for 311

t 6 ll. 3 e 3

IQ(F(t. U)? - Fn(t. u)3)du + O as n + m

42

Now notice that Fn(t, u): S + sp{n1,...,nn} for all t, u e 11,
i.e., the range of Fn is of dimension less than or equal to n for
all t, u E 11. Since 311 locally convex Hausdorf topologies on finite
dimensional spaces are equivalent ([Ru], p. 16) and since there exists

a sequence of simple functions {Fén)(t, u):, k 3 1} such that
fllFfi")(t. u)3 - Fn(t. u)3(lzdu - 0 as k - ..

for each n 3 1 it follows that
I0(f,£"’(t. U)? - Fn(t. U)q>)du + 0 as k + a.

for each n 3 l, t e 11. 3 E 3. Hence there exist simple functions
F“) 6 L2(Q. R) (note that all simple functions are in L2(Q; R ))
such that

IQ(F(k)(t. u)3 - f(t. u)p)du . 0 as k .. ..

forall teR,3€S. Thus we have that foreach 3ES,t€ R,
f(F(k)(t, u)3, W(du)) is a sequence of Gaussian random variables that
coverges in L2(n. F, P) and hence converges in probability. We call
the limit in probability the generalized stochastic integral of F
(with respect to W evaluated at 3) and denote it by

(2.3.10) “F(t, u)3. W(du))
or (suppressing .3) formally by
(2.3.11) [F'(t, u)W(du)

For some Borel set A : I1 we define (as before)

43

(2.3.12) i F'(t. u)W(du) = [IA(u)F'(t, u)W(du)

2.3.6. Properties of the Stochastic Intggral.

By construction it follows that f(F(t, u)-, W(du)) 6 S' P-3.s.
for 311 F e L2(O; 12). Since (2.3.10) represents the limit in probability
of mean zero Gaussian random variables it follows (by [M1], p. 24) that
(2.3.10) defines a mean zero Gaussian random variable. By (2.3.5) it

follows that
B34” Euoh.npwwwn2=muu,npn.+.
and by (2.3.6) it follows that

(2.3.14) Ef(F(t.--U)CP. W(dU)) f (F(S. UN. N(dU)) =(FfQ t. u )3. HS. U)p )dU

Notice that it is possible for F e L2(Q; I!) to have the form
F(t, u) = e(t)w(u) where o(t)3 is non-measurable with respect to dt
and yet W(u)3 is a Lebesgue measurable element of L2(Q; 11). We
will therefore make the convention of calling F measurable if F(t, u)
is jointly measurable in t. and u.

As a final elementary property of generalized stochastic integrals

we give the following analog of Fubini's theorem.

2.3.7. Theorem (Fubini). Let F e L2(Q; R) be such that F(t, u)

is strongly measurable in t (i.e., F is measurable) and for all 3 6 S
(2.3.15) If0(f(t, u)3)dudt < + .
Then

(2.3.17) f[fF'( W(du)]dt= f[fF(t , u )dt]'W(du)

44

3399:, First note that F(t, u) being strongly measurable in t
guarantees the existence of [F(t, u)dt as an element of B(S) ([HP],
p. 85) and (2.3.16) guarantees that it is in fact in L2(Q; 11).
Consider the case when F(t, u) = .3] F(t, j) IAj(Aj e 80), i.e., F

J:
is a simple function. Then, operating formally we obtain

ffF'(t, u)W(du)dt

n
I I F'(t. j)W(Aj)dt
i=1

n
,I (IF(t. j)dt)'N(Aj)
i=1

f[fF(t, u)dt]'W(du)

Now for an arbitrary element F E L2(Q; ll) satisfying the conditions
of the theorem, there exists a sequence of simple functions {Fn(t, u)}

s: L2(Q; R) such that

[Q(Fn(t. u)3 - F(t, u)3)du -» 0

Then by (2.3.16) it follows that
foFn(t, u)dt]'W(du) + jtff(t, u)dt]'W(du) in probability

and
IEIF5(t. U)W(dU)]dt +~f[fF'(t. u)W(du)]dt in probability
So, (2.3.17) follows 0.5.0.

2.3.8. Remark. As a final remark we would like to note that we feel

Ito's definition of the stochastic integral ([I]) is not entirely

45

rigorous. Although, as Ito states, the construction is similar to
Kuo's construction, Sato's theorem, crucial for the argument, was not
known until recently ([53], pp. 299-313). Nevertheless, Ito's ideas
led us to realize the connection between Kuo's construction and Sato's

theorem.

52.4. Stationary Solutions of the Generalized Langevin Equation.

In this section we will give an appropriate interpretation of
the "generalized" Langevin equation and exhibit a solution which is
stationary. By stationary we mean, of course, that the covariance of
the process is of the form C(lt-sl, 3, 3). We will also examine the
relationship between the stationary solutions and the non-stationary
solutions obtained for the restricted Langevin equation of section 1.6.
In order to better understand the general discussion we will begin

with an example that complements example 1.3.1.

2 2

2.4.1. Example. Let 8: L + L be a bounded linear operator; let

{W(A): A 6 80} be a generalized Wiener set process with covariance
given by A(A1 n A 2) <83, 83> defined on a probability space (9, F, P).
Notice that I(_mt J(u)e B(t ")62 L (“B H2 ; ll) (3 > 0) since

NIB I(-~. ”(Ule's 8'“ ">311sz
t
= “Benz I e'28(t‘“)du

= 1183112 (3,) < + -

Therefore, we can define an S'-valued process {V(t): t e 11} by

46

(4. V(t))= (gamut) e'8(t‘“’(4. w(du))

t
(2.4.1) = I (e'8(t'")3. W(du))

-Q

By the remarks of 2.3.6 it follows that (3, V(t)) is a mean zero

Gaussian random variable and

5(3. V(t))(w. V(S))

tAs
= I <3 e-8(t-u)¢’ B e-g(S-u)3>du

tAs
= (3?, 33> I e-B(t + 5-2") dU
(2.4.2) = <83, 83> é%-e-Blt-sl

Thus. {(3. V(t)): t e 11} is simply a stationary Gaussian process
with variance parameter H83U2. Obviously the process {V(t): t e 11}
has all properties discussed in sections 1.3 and 1.4.

Following the argument in [L11] and [L12] we examine the equation
t
(2.4.3) (3. V(t)) - (3. V(S)) = I (-63. V(U))du + (p. W(ts. t]))
s .
This is similar in sturcture to the restricted Langevin equation
t
(1.6.5). Let -3 < s < t < w and consider I (-33, V(u))du.

s
Letting V(u) be as in (2.4.1) we obtain

(-83. V(U))dU

(-B e'8("'”)r. W(dv))du

IS:

t

I

S

t

=I

S

t -B(u-v)

= f (-8 e 3, W(du))du
5

IR.“

on

47

= I (I -s e'8(”'”)¢ du, W(du))
-m S
t t ( )
+ f (f -B e.8 u-v m du, W(dv)) by theorem 2.3.7
5 v
= ? (e‘B‘t'”)¢-e’3(5'v)¢. w<dv))
t -B(t-v)
+ f (8 ¢'@: W(dV))
s
t t

I (e'B‘t'V’a. w<dv)) - ? (e'B(S'V’¢. w(a.)) - I (w. W(du))
S

-Q -Q

= ($9 V(t)) ‘ (¢a V(S)) ' ($9 “([5, t]))

Thus we have shown that (2.4.l) solves (2.4.3)

Using a similar argument we can show that
t -B(t-u)
(2 4.4) (m, V1(t))= é (e v. N(dU))
(a mean zero Gaussian random variable) is (in fact a unique) solution to
t
(2.4.5) (o, V(t)) - (v. V(0)) = g (-Bo. V(S))ds + (m. "([0, t]))

where V(O) a 0. The covariance of (2.4.4) is easily computed as

A5

t
(2.4.6) E(¢, v1(t))(w. V1(s)) = g <3 e'6(t‘”)¢, B e'B(S'“)w>du

showing that V](t) is non-stationary.

The only difference between equations (2.4.3) and (2.4.5) (and
their respective solutions, (2.4.l) and (2.4.4)) is the range of the
time parameter. For (2.4.3) ((2.4.l))t e I? and for (2.4.5)((2.4.4))

48

t 6 12+. But as we have shown (2.4.1) is stationary and (2.4.4) is

non—stationary.

Let Ft = o{(¢, V(S)): -w < s f t, m E S}, t E 1?. Ft is

called the history of the process up to time t. Obviously,
t - o{(q>. W(A)): A 6 80 (-°°. t]. Op 6 S}. t e R . Also. let

‘7"! ‘H
I I

t - a{(cp. W(A)): A e Bo‘n[t, + a»), cp e 3}, t e R. Define by PF+
the projection of L2(Q, F, P) onto the Gaussian subspace M generated
by F: -measurable random variables. That is, PF+ is the projection
of a Gaussian random variable onto the "future starting at time t".
Since we are working with Gaussian spaces, it follows (by [Ml], p. 40)

that

(2.4.7) P + M = E(-IF:)

Then for t i 0

.0
m
+
A
'8
U
<
A
fl
v
V
II

t
EU (e‘3(t'”)cp. maumrg)
t -B(t-u)
(2.4.8) = (g (9 <2. N(dU)) = (<9. V](t))

We call F3 (F0) the pure future (pure past) of the process. Then the
solution of (2.4.5) is the projection onto the pure future of the
stationary process defined by (2.4.1).

We will call equation (2.4.3) ((2.4.5)) the Langevin (restricted
Langevin) equation. We can write (2.4.5) in a more pleasing manner by
letting W(t) denote “([0, t]) and H(O) s 0. By suppressing m

we can express both equations formally by

49

t
V(t) - V(O) = 6 (-8)V(s)ds + W(t) for t 3 0

(2.4.9)
t

V(t)-V(s) = f (-B)V(s)ds + N([s, t]) for -w < s < t < w
s

As per convention we write (2.4.9) in differential form as
(2.4.10) V(dt) = -3V(t)dt + W(dt)

Note that (2.4.10) has the same structure as (1.1.1) - only the symbol
interpretation has changed.

One final comment. The non-stationary solution to (2.4.10)
is the solution with an initial condition of V(O) s 0. If we have
V(O) a V0 6 3' P-a.s. then it is easy to see that the non-stationary

solution is given by

t
(e. V(t)) = (e‘Bte. v0) + g (e‘B‘t'u’e. w(du))

2.4.2. The Generalized Langevin Equation.

Using the methods of example 2.4.1 we are now ready to consider
the general case of the Langevin equation. Assume that l.4.l(a) and
(b) hold. As before, 1et {N(D): D 6 30} be a generalized Wiener set
process with covarince given by A(D1 n 02) <B¢, B¢> defined on a
probability space (a, F, P). Furthermore, we assume that

2 2, .
I(_m’ t](u) Tt-u 6 L (“3°H . ll), i.e., assume

t 2 ° 2
(2.4.11) I “B Tt-u Q“ du = 6 “B Tum“ du < + m

~€D

Therefore the S'-valued process {V(t): t e 12} given by

50
t
(2 4 12) (4. V(t)) = f (Tt_u¢. W(dU))

is well-defined. The random variable (o, V(t)) is Gaussian with

mean zero and

E(¢. V(t))(w. V(S))

tAs
= f <BT

—co

t-uv’ BTS_uw>du

= ( <BTt-(tas)+u¢' BTs-(tas)+u‘”’d”

f <BT
0

j <Biuo, BTS_tw>du if t < s
o

t-s+u¢’ BTuw>du if s < t

(2.4.13) =

which implies that the process is stationary. By the remarks of
sections 1.4, 1.5 and 1.6, it follows that {V(t): t E 12} is a

sample-continuous S'-valued GGP.

2.4.3. Theorem. The process V(t) given by (2.4.12) is a solution to
‘ t

(2 4 14) (o. V(t)) - (m. V(S)) = I (As. V(U))du + (o. W(ES. t]))
s

(the generalized Langevin equation) for -m < s < t < + m.

Proof. Letting V(u) be as in (2.4.12) we have
t .
I (Am. V(U))du

S

(Tu_vA¢, N(dv))du

l‘:

ll
m‘fifl'

O

51

I‘m

(Tu_vA¢, W(du))du

II
M‘Hfl

(T

+
mk-sfi

40%:

u_vAm, W(du))du

s t

= I (I Tu_vA¢ du, W(dv))
-m s

+

( T _ Am du, W(du)) by theorem 2.3.7

UV

cg‘fi

mat-r

S
= I (1t-.¢ - Ts_v4. W(du))
t
+ I (Tt_ch'CPs N(dVH
S
t S t
f (Tt_v¢s W(dv)) ' f (Ts-V¢, N(dV)) ' I ($9 W(dv))
S

(4. V(t)) - (m. V(S)) - (w. W(ES. t])) Q.E.0.

Once again a similar argument gives us an immediate corollary.

2.4.4. Corollary. The sample-continuous S'-va1ued GGP {V](t): t 3 0}
defined by

1:
(2.4.15) (4. V](t)) = (Ttv. V0) + 6 (Tt_u@. W(dU))
where V](0) a V0 6 S' is a unique solution to

t

(2.4.16) (4. V(t)) - (m. V(0)) = 6 (A4. V(S))ds + (m. W(t))
where W(t) denotes N([O, t]), t 3 0 (and N(O) s 0). Furthermore

tAs
(2.4.17) E(¢, V](t))(¢, V](s)) = 6 <BTt-uv’ BTS_uw>du

52

We call (2.4.16) the restricted Langevin equation. By (2.4.17) it

4.

follows that V1(t) is a non-stationary process. Let Ft’ Ft and

PF* be as defined in example 2.4.1. Then
t

2.4.5. Lemma. PFO [(m, V(t))J = (m. V](t)) for all t 3 0. Q E 3.

Proof. Obvious. 0.5.0.

Once again we have that the solution of the restricted Langevin equation
is simply the projection onto the pure future of the stationary solution
of the generalized Langevin equation.

Generally we will write (2.4.14) (and(2.4.16)) in the formal

differential form
(2.4.18) V(dt) = A' V(t)dt + W(dt)

Solutions of (2.4.18) will be called generalized O.U. velocity processes.

Note that this equation (as in example 2.4.1) has the same structure

as (1.6.8) and (1.1.1) - only the symbol interpretation is different.
Finally, note that (2.4.15) and (1.6.9) as solutions to the

restricted Langevin equation are the same process (the covariances

are equal). This is not surprising in view of the fact that (1.6.9)

is simply the "integration-by-parts" form of (2.4.15).

2.4.6. Remarks. (a) Conditon (2.4.11) implies that “T decay

til
at some exponential rate as t +.¢; this is not unduly restrictive in
view of the one-dimensional case.

(b) It is easily seen that both processes V(t) and V1(t) are Markov.
(c) Using an argument similar to that in section 1.6 one can show

that EYS t(€P)YO,T(¢) = X(ts. t] n [0. 100(4). iv) for

53

-m < s < t < m, -w < a < t < m where Ys t(4) = (4. V(t)) -

t
(4. V(S)) - I (A4. V(U))du
S

(d) Uniqueness of the solution in corollary 2.4.4 is immediate by the
results of section 1.6 and the fact that (1.6.9) and (2.4.15)

have the same covariance.

52.5. The Generalized Displacement Process.

In section 2.4 we discussed the generalized O.U. velocity process
{V(t): t e 12} given by (2.4.12) (the stationary solution of the
Langevin equation). In this section we will define the corresponding
"displacement" process. Furthermore, we show that the process describing
the displacement of a Brownian particle converges (in distribution)
to a non-interactive space-time process (namely, a generalized Wiener
process) when the surrounding medium of the particle undergoes certain
physical changes.

In any physical motion velocity is defined to be the time
derivative of the displacement or, conversely, displacement is the
integral of the velocity. Similarly we define an 3'-valued process

{X(t): t 6 12}, called the generalized O.U. displacement process, by
t

(2 5-1) (4. X(t)) - (4. X(S)) = f (4. V(U))du for -w < s < t < w
5

Note that the integral in (2.5.1) exists since V(t) is sample-con-

tinuous. Obviously (4, X(t)) is a mean zero Gaussian random variable.
Although (2.5.1) readily suggests that the explicit form of

X(t) is given by F (4, V(u))du this integral may not exist. In

order to simplify our discussion we will consider only the pure-future

projection of this integral; i.e.,

54

t
(2.5.2) (4, X(t)) - (4, X0) = é (4. V(S))ds, t 3 0, X(O) 5 X0 E S' P-a.s.

where V(t) is the non-stationary solution to the restricted Langevin
equation; i.e., V(t) is given by (2.4.15). We will further assume

that V(O) 2 X(O) a 0. So, substituting into (2.5.2) we obtain (formally)
t 5
(2.5.3) X(t) = j j T' N(du)ds
o o 5‘“
Note that
t 2 2 2 2
g “B I Ts-u4 dsu du 5 5 t H8“ H4“ < + 4
u

t
2 2.
so IE0,t](u) i Ts-u¢ ds 6 L (“B-fl . 11). Thus, by theorem 2.3.7 we

obtain (formally)

X(t) = ? T' -u N(du)ds
0

S

(2.5.4) =

C‘sd’ Oksd’

t
(I Ts-u ds)l N(du)
u

Thus, the covariance of X(t) is easily seen to be

tAs t s
(2.5.5) E(o, X(t)(p, x(s)) = I <3 I T 4dv, B j T 4dv>du
0 u V‘U u V‘U

Assume that there exists a bounded linear operator 0: L2 + L2

such that OA'= B. Let {An: n 3 0} be a family of bounded linear
operators on S such that each admits a self-adjoint extension Ah
to L2 that generates a strongly continuous contraction semi-group

{T£n): t 3 O} and AC): A. Let (3"; n 3 0} be a family of operators

2 _ -
on L such that B0 - B and D An - 3n“
and linear on L2. Let {Xn(t): t 3 0} denote the displacement process

Obviously Bn is bounded

corresponding to the covariance

55

(2 5 3) )AS <3 } T‘”) dv B s 1(") w dv>du
' ' 0 n u v-U ¢ ’ n i V-u
2.5.1. Theorem. If <Tén) 4. ¢> 4-0 as n +‘4 for each t 3 O

and for all 416 3, then the process Xn converges in distribution

to a generalized Wiener process with covariance (tas) <0¢, 04>.

Erggf, Note first that Tfin) 3 0 since A' is self-adjoint (this

follows by the spectral mapping theorem [Ru], pp. 305 ff.) and thus

|<T£n) 4. 4>l2 f <Tén) 4 , ¢><Tén) w, n> ([Be], p. 149). So, <Té") o, 4> + 0
for each t 3 O, and for all 4, n e S as n +.m. Since Xn is a

Gaussian process it suffices to show that E(¢, Xn(t))(¢, Xn(s))a.(tas)

<04, 04>. By (2.5.3) we obtain

5(4. Xn(t))(4. xn(s))

tAs t-u

s-u
= 5 <Bn é TE") ? dv, Bn f T5") w dv>du
0
tAs t-u‘_ s-u _
= 6 <0 6 AnTgn) 4 dv, D g AnTsn) 4 dv>du
tAs
= (n) (n)
g <D(Tt_u P ' ¢)9 D(TS‘U 4 ' 4)>dU

tAs
n
= I E<DT£_3 4. Dtéffl 4> - <0T§33 4. w>

- <Dcp, mé’ja up + 43.9, 0.)»de

tAs
+ g <04, Ow>du. by the dominated convergence theorem

= (tAs) <04. 04> 0.5.0.

56

2.5.2. BEEEEE: Although a "physical" interpretation of the above
theorem is not apparent we can, by examining the one-dimensional semi-
group case, arrive at a satisfactory interpretation. Let us begin by
"physically interpreting" the Langevin equation (2.4.18). This equation
gives us a description of the velocity of a microscopic particle in
liquid suspension (a so-called "Brownian particle") in the absence

of an external force field. The term A'V(t)dt represents the momentum
loss due to frictional forces (viscous forces) and W(dt) represents

the momentum transfer due to molecular bombardment. Hueristically,

the condition that T£n)i+ 0 corresponds to the increase of frictional
resistance of the surrounding medium (i.e., an "increase" in An).

But we allow the friction to increase only at a rate proportional to

an increase in the variance of momentum transfer due to molecular
bombardment (i.e., an "increase" in B"). The constant of proportionality
corresponds to 0. (See e.g. [Br]. PP. 347-349 and [N], pp. 53-61.)
Allowing the frictional resistance and the momentum transfer to "increase
to infinity" we would expect that our displacement diffusion should
converge to a process (in particular, a Wiener process) where there is

no interaction between time and space parameters. This is precisely
what theorem 2.5.1 illustrates.

More importantly however, this theorem gives us a dynamical
construction of a space-time Wiener process (Brownian motion) on an
infinite-dimensional space. This follows in the spirit of the original
work of Ornstein, Uhlenbeck and Wang (see [N], pp. 53-61 and [0 U 1.
pp. 93-111 and [WU], pp. 113-132). This also serves to give us a basis

on which to consider the quantum Langevin equation, as described in

57

[FKM] and [BK], for space-time processes.
As a final note we mention that theorem 2.5.1 is a generalization
of theorem 9.5 in [N] (p. 61) where only one-dimensional processes are

considered.

CHAPTER III
GENERALIZED EVOLUTION EQUATIONS AND THE
CHEMICAL-REACTION MODEL

53.1. Introduction.
Recall that in Chapter II we specified the restricted Langevin

equation as
t
(3.1.1) V(t) - V(O) = 5 A'V(s)ds + W(t); V(O) a V0 6 S' P-a.s.

In this chapter we consider two generalizations of this equation. The
first involves the "non-homogeneous" or "evolution" equation. Examining
this type of equation answers the question of what would happen if the
semi-group generator A were allowed to vary (in some suitably "smooth"

manner) with t. In other words, consider equations of the form

6 S' P-a.s.

t
(3.1.2) V(t) - V(O) = g A'(s)V(s)ds + W(t); V(O) a V0

The second generalization concerns the "evolution equation with
a linear driving term", what we will call the "chemical-reaction"

equation. By this we mean equations of the form
t t
(3.1.3) V(t) - V(O) = f A'(s)V(s)ds + f o'(s)W(ds); V(O) 5 VO 6 S' P-a.s.
' O 0

Although proving the existance of a solution for equation (3.1.3)

is quite easy the uniqueness of this solution is more difficult to prove

58

59

than in the cases of the Langevin and "evolution" equation. As we shall
see we need be more subtle than in the previous cases.

Both of these equations provide a means of modelling linear
stochastic phenomena with a disturbed nature, for example, turbulence
and certain chemical reactions with a deterministic linear driving
term (hence their respective names). For more details one should refer
to [0]. [ACK] and its bibliography and [LP].

We also give an appropriate definition of the infinitesimal
generator of the "chemical-reaction" process (i.e., the solution of
(3.1.3)). As we shall see, this definition is motivated by and consistent
with the form of the infinitesimal generator of a finite-dimensional
process.

Finally, we will give the appropriate analog of Ito's formula

([F]. Pp. 78-85) for the "chemical-reaction" process.

53.2. Generalized Stochastic Evolution Equations.
In this section we examine the solution of (restricted) stochastic

differential equations of the form

(3.2.1) V(dt) = A'(t)V(t)dt + W(dt); V(O) a V e S' P-a.s.

O

which we will call generalized stochastic evolution equations, where

A(-) varies in a suitably "smooth" manner and for each t 3 O, A(t)
generates a semi-group. We begin with some basic definitions and theorems
of evolution operators; further details can be found in the references

cited.

3.2.1. Definition ([60], pp. 160, 161). Let L be a Banach space and

B(L) the bounded linear operators mapping L to L. Let

60

0 = {(t,s): o f s 5 t < m}. An evolution operator is a map U(o. ')

 

03+ B(L) such that
(a) U(t, s)U(s, T) = U(t, T) for O f T f s f t < 4 and
(b) U(t, t) = I for all t 3 0.
We say that U(-, .) is strongly continuous if U(o, .) f
is continuous on 0 in the norm-topology of L for each f e L; U(-, -)
is a contraction if ”U(t, s)fl|L f flfflL for each (t, s) e 0.
Let A be the collection of all generators of strongly continuous
contraction semi-groups of bounded linear operators on L to L. For
A 6 A let 0(A) denote the domain of A and {T(t, A): t 3 O} the

corresponding semi-group.

3.2.2. Theorem ([60], pp. 159-160). Let A(-): I2+ +-A. Suppose that
T(s, A(t)) and T(o, A(T)) commute-for all t, s, o, T 6 11+. Suppose
there is a dense linear manifold v c L such that D : 0(A(t)) for
each t 3 0 and A(t)f + A(t)f as t + T for each f e D, T 3 0.

Let

5 = n [{lim n 0(A(t))} n {g e L: 11111 A(t)g = A(t)9}1
1230 60+ It-tl<€ 1+1:

Then 5 G D and there exists a strongly continuous contraction evolution

operator U(-, -) such that for f E D and O f s 5 t < a

(3.2.2) 5‘"? U(t, s)f = A(t)U(t, s)f = U(t, s)A(t)f

3.2.3. Theorem ([60]. pp. 151-153). Let A(-): R++A and let

RA(b. a) = T(ant. A(rn))--oi(a,(t), A(r1)>

61

where O f a f b < m, A: a = t <---< tn = b is a partition of

O
[8. b]. t. 6 Et. , t.], 1 < i < n, lAl = max |A.tl = max |t.—t. |.
1 1-1 1 - - 15,5" 1 lgifn 1 1—1
If Ilim RA(b, a)f exists for all f e L, 0 f a 5 b < a, this limit
A +0

is denoted by U(b, a)f and U(-, .) is an evolution operator in
B(L). Moreover, if T(s, A(t)), T(o, A(T)) commute for all
s,t, o, t 6 12+, U(o, -) is a strongly continuous contraction evolution
operator.
3.2.4. Definition ([60], p. 163). If 1im R (b, a)f = U(b, a)f

IA4+0 A
exists for all f e L, then A(-) is cal ed the propagator of U(-, .)

 

Let {A(t): t 3 0} c B(S) be such that for each t 3 O, A(t)
satisfies conditon l.4.l(a). Furthermore, let A(-) be the propagator
of a strongly continuous contraction evolution operator U(-, -) on L2
such that (3.2.2) is true. Let {W(t): t 3 0} be a generalized Wiener
(set) process with covariance (t A s) <B¢, Bw> where B satisfies

l.4.l(b) (t, s 3 0). Notice that for all t 3 0. 4 6 S.
t 2 2 2
A “BU(t. 514“ ds 5 t H3“ "4“ < + 4

which implies that Ito’t](s)U(t,s) e L2(flB-fl2; 12). Thus we can define
a mean zero Gaussian random variable by
t
(3.2.3) (4. V(t)) = g (U(t.5)4. W(dS))
Obviously {V(t)} t 3 0} defines an S'-valued GGP whose covariance
is given by

tAs
(3.2 4) E(4. V(t))(w. V(S)) = A <BU(t.v)4. BU(S.v)4>dv

62

3.2.5. Theorem. If {V(t): t 3 O} is defined by (3.2.3) then it is
a unique solution to (3.2.1) with V(O) s O that can be realized in

C([0, 4), 3'). We call such a process a generalized evolution velocity

 

EFOCESS .

Proof. To show that V(t) is a solution to (3.2.1) consider

(A(5) 4. V(S) ))d5 I I (U(s. v)A(S)4. W(dv))ds

I
0
t t
=g (I U(s, v)A(s)4 ds, N(dv)) by theorem 2.3.7
t
6 (U(t.v)4 -4. W(dt)) by (3.2.2)

(4. V(t)) - (4. W(t))

Now, by an argument similar to that of lemma 1.6.4 one can show that
(4, V(t))- I (A(s)4, V(S))ds has covariance (t A s) <84, 34>. Thus
by corollarges 1.5.3, 1.6.5, 1.6.6 it follows that V(t) can be realized
in C(CO, 4), S') and is a unique solution to (3.2.1). Q.E.0.

The general solution to (3.2.1) is easily seen to be given by

t
(3.2.5) (4. V(t)) = (U(t, 0)4. v0) + 5 Mt. s15. w<ds))

Similar to before, we define the generalized evolution displacement

process by
. t
(3.2.6) (4. X(t)) = (4, X0) + 6 (4. V(S))ds

where X(O): X0 6 S‘ P- -a. s.

63

53.3. The Chemical-Reaction Equation.

In this section we discuss the second generalization mentioned
in section 3.1; that is, we solve the evolution equation (3.2.1) with
a "linear driving term" 4(5), where I[0,t](s)°(s) e L2(flB-u2; 12).

Such equations have the form

. t t
(3 3.1) (4. V(t))-(4. V0) = 5 (A(S)4. V(S))ds + g (4(514. W(dS))

where V(O) a V0 6 S' P-a.s.. Following [ACK] we call such equations

generalized stochastic chemical-reaction equations.

 

We will continue to observe all assumptions and notation of
section 3.2. Since I[0 t](s)4(s) E L2(uB-fl2; 12) the integral
‘ t

(3.3-2) 6 (4(514. W(dS))
is well-defined for all t 3 O, 4 e 3. Also assume that IE0 t](s)4(s)U(t,s)
e flue-((2; R). Then
t
(3.3-3) (4. V(t)) = A (4(S)U(t.S)4. W(dS))
is a well-defined mean zero Gaussian random variable for all t 3 0,
4 e 3. Furthermore, {V(t): t 3 O} is an S'-valued GGP with covariance

tAs
(3.3 4) 5(4. V(t))(w. V(S)) = g <Bo(v)U(t.v)4. B¢(v)U(s.v)w>dv

3.3.1. Theorem. The process V(t) defined by (3.3.3) is a solution

to (3.3.1) with .V(O) o P-a.s..

Egggfi, Substituting (3.3.3) for V(t) into (3.3.1) we obtain for each

9 E S

64

ts
(A(S)4. V(S))ds = g 6 (4(v)U(s.v)A(S)4. N(dv))ds

0g!” O‘sfl

t
(4(v) I U(S.v)A(S)4 ds. W(dv))

by theorem 2.3.7 and lemma 1.6.2

t
g (4(v)(U(t.v)4 - 4). W(dv))

t
(4. V(t)) - é (4(v)4. W(dv)) Q.E.D.

If V(O) 2 V0 6 S' P-a.s., then a solution of (3.3.1) is given by
t
(3.3-5) (4. V(t)) = (U(t.0)4. V0) + 6 (4(v)U(t.v)4. N(dv))
We will refer to solutions of (3.3.1) as chemical-reaction processes.
To prove the uniqueness of (3.3.5) as a solution to (3.3.1)
we must rely on techniques that are different than those used in the
evolution equation since (3.3.2) is no longer a Wiener process. We

defer the proof of uniqueness to the appendix of this chapter.

53.4. Infinitesimal Generators of GGP's and Ito‘s Formula.

 

It is well-known that under appropriate conditons the infinitesimal

generator fOr an n-dimensional stochastic process is given by the operator

 

( ) n ( ) 32 E ( ) a
3040] L = 13 a.- t,‘ + b. t,o -—
t 1,§=1 1] axiaxj j=1 J axj

on Cl’z([0, on) x R", R) (see e.g. [F]. p. 117 or [SV]. pp. 82-152).
In this section we give an appropriate definition of the infinitesimal
generator of infinite-dimensional GGP's. (The most general GGP we will

consider is the chemical-reaction process of section 3.3.) To maintain

65

a consistent theory of stochastic processes we desire that if a GGP
is restricted in such a manner as to give a finite-dimensional process
then the "restriction" of the generator of the GGP to finite dimensions
has the form (3.4.1). It is in fact this desire that will lead us to
the definition of the generator for a GGP. We shall begin by examining
a few consequences of the Minlos-Bochner and Sato theorems.

Let Q be a continuous bilinear covariance on S x S. Then by
the Minlos-Bochner theorem there exists a (centered) Gaussian-Radon
measure y on 3' and by Sato's theorem there exist "j e S, g. E S'

J
such that

y = E (nj. yitj y-a.e.(y) (y e 3')
i=1

(3.4.2) and

((Ps') ~ "(0: Q((P))

In particular, if {W(A): A 6 80} is a generalized Wiener (set) process
with covariance A(A n B)Q(4, 4) then
(3.4.3) W(A) = I .(n-. W(A))g. for a11 A e 50
j:] J J
Also, by Sato's theorem it follows that ("j’°) is an independent

sequence of N(O, 1) random variables. Thus

r1 if j = k

(3°4-4) EY(nj3°)(nks') = Ojk =

 

0 if j f k
K

Thus by (3.4.2) it follows that Q(nj, nk) = ij and so

66

(3.4.5) E(nj. W(A))(nk. w(B)) = .(A n 315,,

which implies that (n3. W(A)) and (nk, W(B)) are independent for
all A, B 6 80 when j f k. Obviously (”j’ W(A)) ~ N(O, A(A)).

Furthermore
5(4. w(A))(4. w(B))
= j i=1 (4. tj)(4. Ek)E(nj. W(A))(nk. N(B)) by (3.4.3)
= £31 (4. 53m, gk)6jkA(A n a) by (3.4.5)
.—. A(A n B) 321 (4. ej)(4. 53-); i.e..
(3.4.6) 0(4.w) = E (4. ij)(4. Ej) for a11 4. w e s

i=1
Let A(t) be as in section 3.2, i.e., let A(-) propogate a
strongly continuous contraction evolution operator U(-, °) on L2.
2 2, . .
Let I[0,t](s)¢(5)’ Ico,t](s)o(s)U(t,s) e L ("B-H . 11). We will descr1be
the Uinfinitesimal generator" for chemical-reaction processes of the
form (3.3.5). We begin by describing a "canonical finite-dimensional",

form for V(t) and deriving its generator.

Define an $'-valued GGP v"(t) by

t
(4. v"(t)) = (U(t.0)4. v0) + g (4(v)U(t.v)4. P5 w(a.))

. t m

(U(t.0)4. v0) + g (4(v)U(t.v)4. Pgtjgl (nj. W(dv))€j])
n t

(U(t90)¢9 V0) + z I (¢(V)U(t,V)Q, 5°)(njs N(dV))

3'10 3

(3.4.7)

67

(See section 2.3 for more details on the operator Pa.) Note that the
second equality of (3.4.7) follows by (3.4.3). It is easy to see that
Vn(t) is the solution to

t t
(3.4.8) (4. v”<t)) - (4. v"<o11 = g (A(S)4. v”(s))ds + g (4(S)4. w“(ds))

where

n
n _ . -
(3.4-9) W (A) - Pn W(A) - jg) (nj. W(A))Ej

and V"(0) a V(O) 5 V0 6 S‘ P-a.s.. (The proof is similar to that of
theorem 3.3.1; the uniqueness argument is given in the appendix.) By

(3.4.3) it follows that

(3.4.10) (4. v"(t)) 4 (4. V(t)) for a11 t 3 o, 4 e 3
Furthermore, by Sato's theorem,

(3.4.11) (anJ, Wu.» 4 (cp, V(t)) for all t 3 o, o e 3
Now, by (3.4.8) we have

((an. v"<dt)) = (A(t)Pn4. v”(t))dt + (4(t)Pn4. w"(dt))

(3.4.12) (

 

Lv"(0) 2 v0 P-a.s.

But, since P: = P" ((Prg)2 = P5) it follows by (3.4 12) and (3.4.9)

that

(Pn4. v"(t)) = (A(t)Pn4. ng"(t)>at + (4(t)Pn4. w"(dt))
(3.4.13)

Vn(0) 5 V0 P-a.s.

68

Now, by applying the definitions of Pn’ PA, W" we obtain
n n n n n
X ($.Ek)(nk. V (dt)) g kg] (Qsfik) jg] (njs V (t))(A(t)nk.€j)dt

n
.2 (9(t)nk.€j)(nj. W(dt)). Vn(0) 2 V0 6 S' P-a.s.

n
+ Z (4.5k)
k=1 J=1

(3.4.14)

This is the form we want since all terms are in terms of the Satoe

"coordinate" vectors, "j and gj.

In order to simplify our equations we introduce the following notation:

En = ($19--~9¢n) = ((¢3§])9---9(¢9§n))
y?(t) = (v?(t)...,v3(t))T = ((n].Vn(t)).-o-.(nn.Vn(t))T

(T denotes transpose)
n
n . + n . n =
bk(t,x). R x R -> R , bk(t,x) J.glx‘j(A(t)nk,gj), l 5 k 5 n

3.4.15
( ) 9p(t.x) = (b?(tix)’°°°’b2(t’x))T

0:.(t): 12* .-ii. .Q.(t) = (4(tInk.€j),1 g j.k 5 n

 

 

J J
r
.410.) 01).“;
o"(t) = . .
L434 4.4
J

wF(A) é <w1(A).....wn(A)I = ((n].W(A))..--.(nn.W(A)))T

(Note that (nk, W(A)) = (”k’ w"(A)) for 1 5 k 5 n.) Then (3.4.14)

can be written as

69

(3.4.15) 4n-1n(dt) = gn-b"(t, 1"(t))dt + gn-on(t)-Wn(dt)
v”(0) 5 v0 5 s' P-a.s.

We will call the n-dimensional (ll-valued) diffusion process y?(t)

defined by the n-dimensional system of stochastic differential equations
y?(dt) = 99(t. y?(t))dt + a"(t)yP<dt)

(3.4.17) y?(0)

((n].Vn(0)).--..(nn.Vn(0)))
= ((n],V0),.. -a(n “’VO)) _3

where V"(O) a V e S' P-a.s.

O

the canonical n-dimensional form of V(t). It is easily seen that

!p(t) has infinitesimal generator Ln on 02(12") given by

t
= 5 Z a" W(t) I b"(t -) ~3—-
j- _] axa axj jg] j axj
(3.4.18) wheren

a?j(t)= k214?k(t)o§k(t)

(see e.g. [F], pp. 116-117).

This is not quite what we want. Since V"(t) is an S'-va1ued
process we would like to define its "generator" for a class of functions
on S'. This is easily accomplished by considering the following de-

finition.

3.4.1. Definition. A function f: S' +-12 is called n-tame (n 3 l)

 

if there exists a function 1": R"

+-12 such that f(y) = f((n1,y),...,
(nn.y)) for all y e S'. A function is called tame if it is n-tame

for some 1 5 n < + m.

70

We will say that f: s' + 12 is in n-c2(s') if f is n-tame and
f 5 52(12"). Then define for 1 5 k 5 n. y e 3'
(3.4-19) b (t.Y) = Z (n-.Y)(A(t)n .E-)

k j=1 J k J

Now define [2 f for f e n-c2(s') by

n
(3.4.20) (t2 mm = ti g 1 r: ,x f(<n,.y).. ...(nn.y))
. = J
n .
+ j; 33% .y)-—— ,xj f (1.13) .-.(nn..Y))

3.4.2. Definition. The operators :2 on n-Cz($') (n 3 l) are called

the finite-dimensional infinitesimal generators of V(t).

 

Let u" be the unique measure on C([0, 4); II") that solves

the Martingale Probelm corresponding to Ln (which we know to exist

t
by [SV], pp. 139-145). Let Rn = sp{n],...,nn} : s and let R: c 3'
denote the annihilator of Rn’ then “n can be viewed as a measure

on Cn = C([0, 4), S'/R;). Letting C = C([0, 4), S') we have by the
obvious interpretation that Cn s: Cn+l C C for all n 3 1. Let

n denote the canonical projections from C onto Cn’ Cn onto

n’ 1rn,m

Cm (n > m 3 1) respectively. Then {Cm; "n,m: l f m < n < + a} 15
a projective family with projective limit C. Furthermore the measures

0 0 _ '1
“n satisfy the consistency conthion “m - “n o "n,m'

3.4.3. Lemna. C is a Suslin space.
Proof. Since 3' is Suslin there exists a Polish space X and a

continuous l-l map 5: X +IS'. It is easy to see that C([O, 4), X)
is Polish. Define E: C([0, 4), X) + C by (8 f)(t) = p(f(t)). then

71

5 is continuous and l-l. Hence C is Suslin. Q.E.0.
Therefore by the Prokhorov consistency theorem for Suslin spaces

([Sc], pp. 74-82 and pp. 288-295) there exists a unique measure 0 on

C such that u" = u 0 n31 for all n 3 1. Of course this measure

is precisely the measure that would be obtained by the methods of

1.4.3 since on finite-dimensional spaces the measures conincide.

3.4.4. Definition. We call the system ([2: n 3 1} the infinitesimal

 

generator of the GGP V(t) with associated u if u o n;] is the
unique solution to the Martingale Problem corresponding to L2.
Clearly this definition is consistent with the usual definition of the
infinitesimal generator of a lln-valued process.

It is also quite easy to obtain a result analogous to the usual
Ito formula. Once again consider the canonical n-dimensional form of

V(t) - equation (3.4.17). It follows immediately that for g e
C1’2(R+ x R", R), g(t, _V_n(t)) has a stochastic differential given by

n _ n §__
dg(t. y_(t)) - (Lt + at)o1t. ("(t114t
(3 4 21) + E _2_.g(. v"(t))o" (t)w (dt)
. o .i -=" 3X. , — ij J
.J 1
(see [F]. PP. 90-92). We can make the same transformation here as
we did in (3.4.20). If g(t,y): R+ x s' + R is such that

g(-.y) (- c‘(R+, R) and g(t.-) e n-C2(S', R) then
dgit.y?(t)) = [(52 + §%9§(t.(n].Y).....(nn.Y))]dt

11
(3.4.22) + i§3=i 533-§(t.<n].y).....(nn.y))ogj(t)wj(dt)

72

There is another way to view the Ito formula though that gives
a result more similar to that in [ACK]. Applying (3.4.3) to (3.4.8)

we have

n
(3.4.23) (4. v"(dt)) = (A(t)4. v"(t))dt + .Z](¢(t)4.€
J:

Then (by [F]. pp. 90-92) for g e c"2(R+ x R, R) we have

j)(nj.N(dt))

d9(t.(4.V"(t))) = [gt-9(5. (4.V"(t))) + (A(t)4. v"(t)) §§-g(t.(o.v"(t1))
32 n
+ % ;_2'g(t9 (@5V (t)))

I (4(t)4.5.) Jdt
x =1 3

(3.4.24) + §§-g(t. (4.V"(t)))

J

“M: L:
._a

(9(t)¢.€j)(njs W(dt))

But by (3.4.6)

2 (4(t)¢.§j)2 = Q(4(t)4)
i=1

and by (3.4.3)

co

2 (4(t)4.ej)(nj. W(dt)) = (4(t)4. W(dt))
j=l

it follows from (3.4.10) that

dMMewmww%Mnmvm0+mumvmr§4nmum»)

2
+ t -3§-9(t.(4. V(t)))Q(4(t)4)Jdt

3X

44%) +§dumvm0mmemmn

(see equation 2.10, p. 5 in [ACK]).

73

3.4.5. Remarks.

(a)

(b)

In this chapter we have defined the concept of the infinitesimal
generator for the general non-homogeneous (i.e., chemical-reaction)
case. In the homogeneous case (Chapter II) with a particular form
of A in the Langevin equation we can get analogs of the Fokker-
Planck equations. The solution of these equations will give a
consistent family of transition functions. Then using the fact
that 3' is Suslin one can piece together these transition functions
to obtain a transition function on 3'. In subsequent study we
shall try to relate this construction in the special case that

A = «:3 (A the Laplacian) to that in [Mi].

It is also possible to study the Feynmann-Kac formula in our
context. This would require, however, development of stochastic
integrals with respect to non-anticipating functionals. We have
made some progress in this direction and intend to complete it in

future work.

APPENDIX

APPENDIX

§A.l. Appendix: Uniqueness of the Solution of the Chemical-Reaction
Equation.

Let Q be a continuous bilinear covariance on s x s and
H(Q) the completion of the space S/ker(Q) under the norm 03(4,4).
Then H(Q) is a Hilbert space with inner product Q(4,4). Here we
identify the equivalence class with the elements in it.

Let {X(t): t 3 0} be an $'-valued GGP such that X(O) 5 O
P-a.s., (4, X(t)) is a martingale for each~ 4 E 3 and for
I[o,t](s)4(s) e L2(Q; ll), (4, X(t))2 - Z Q(4(u)4)du is a martingale
for each 4 6 3.

. tAS
A.1-l. Lemma. E(4. X(t))(w. X(S)) = 6 Q(4(u)4. 4(u)4)du.

2329:, Let F .t 3 0 denote the o-fields with respect to which

t
t
(4, X(t)), (4, X(t))2 - f Q(o(u)4)du are martingales. Note first
0 .

that for all t 3 O,

E(4. X(t)) = E{E[(4. X(t))lFOJI = E(4. X(0)) = 0

and

E(4. X(t))2 = E{E[(4. X(t))leOJ}

t
44.nmfi+6044404

t
6 Q(4(U)4)du

74

75
and for O f s < t < m

Hexunw.041=4n4.u4xwxenpgi

no.04nuaxnnngi

E(4. x(s))(w. X(S))
Therefore
44.nw)-w.uwnz

=Ew-4.04F

and

m4.uo)-w.nwnz

=Ew.nwfi+eo.uofi-zs4.04meen

Hence the lemma follows. Q.E.0.

2 .
It follows by I[0,t](u)¢(u) e L (Q. 11) that Ito’t](-)4(-)4 E

L2(R+, 14(0)) for a11 t 3 o, 4 e 3. Since

E(4. x(t))(p. x(s)) = <1[0,t3(°)5(-)4. Ico.s1(')"’(°)> (2(0“, 11(0))

tAs
= 5 Q(¢(U)4. 4(U)4)du

we get for k(-) e EEIIEO t1(«)4(-)4: t 3 O, 4 6 3} (where the closure

76

is in L2(12+, H(Q))) that the class of functions

t
K = {fk(t. 4) = g Q(¢(U)4. k(U))du= t 3 0. 4 e S}

is the RKHS of the process {X(t): t 3 0}. (see [Ml], pp. 29-30).

Let n be the isomorphism of K onto the Gaussian subspace
generated by {(4, X(t)): t 3 O, 4 e 3} (see the remarks of 1.2.9).
Let K‘ be the orthogonal complement of K in L2(lt+, H(Q)). Denote
by if the extension of 1. to L2(R+, H(C)) such that 370(1) e
's'p'iicp. X(t)): t 3 o, 4 e 51*. Define (4. W(t)) by

(4. W(t)) = T(I[0,t](-)I4)

where I is the identity on H(Q). Then {W(t): t 3 O} is the

generalized Wiener process and

t
F(I[0,t](-)4(-)4)= (I) (4(U)4. W(du))

That is

t
(4. X(t)) = 5 (4(U)4. N(dU))

for some generalized Wiener process W. In fact, a similar argument
shows that given any GGP Z(t) with the covariance

tAs
é Q(U(t. V)¢(v)4. U(S. V)4(VI4)dv

there exists a generalized Wiener process W(t) such that
t
($9 Z(t)) = a (U(t, V)¢(V)¢s W(dV))

Hence the solution to (3.3.1) (and (3.4.8)) is unique (see
[SV]. PP. 99-110).

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