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Diffusion approximation for
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Alla Sikorskii

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Ph.D Statistics
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A.V. Skorokhod

Ma 25 2000
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DIFFUSION APPROXIMATION FOR SOLUTIONS OF PERTURBED
DIFFERENTIAL EQUATIONS
By

Alla Sikorskii

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

2000

ABSTRACT
DIFFUSION APPROXIMATION FOR SOLUTIONS OF PERTURBED
DIFFERENTIAL EQUATIONS
By
Alla Sikorskii

We consider the operator differential equation perturbed by a fast Markov process:

in a separable Hilbert space H. Here y is an ergodic jump Markov process in phase
space Y satisfying some mixing conditions and {A(y), y E Y} is a family of closed
linear operators. We study the asymptotic behavior of the distributions of u.(t/e).
For the case when the operators A(y) commute, Salehi and Skorokhod (1996) proved
that the distributions of u€(t/e) asymptotically coincide with the distributions of some
Gaussian random field with independent increments.

We do not assume that the operators A(y) commute, but we impose some condi-
tions on the structure of these operators. We study the asymptotic behavior of the
stochastic process z€(t) : e“‘/iu€(t), where A = fA(y)p(dy), and p(-) is the ergodic
distribution of the Markov process y(t), t 2 0. We prove that the stochastic process
z€(t/e) converges weakly as e —+ 0 to a diffusion process 2(t), t 2 0, which is de-

scribed using its generator. The proof is based on the theorem on weak convergence

of H -valued stochastic processes to a diffusion process.

To my family

iii

ACKNOWLEDGEMENTS

I would like to express my gratitude to my dissertation advisor Dr. Skorokhod for
everything I have learned from him over the past ten years starting with my first
course on the theory of the stochastic processes and all the way up to advanced
seminars on stochastic analysis and ergodic theory. Many thanks to the members of
the advisory committee: Dr. Salehi who has always been very supportive of me, Dr.
Levental whose enthusiasm about research is so contagious and Dr. LePage whose
optimism enlightened my stay at MSU. I would like to thank Dr. Erickson for the
great teaching experience I shared with him and Dr. Gilliland from whom I learned
a lot about statistics and how to teach it at the undergraduate level as well as many
other things. I’m grateful to Alex White who always helped me when I needed help.

I give thanks to my parents who always encouraged my learning. I couldn’t have
completed this work if it wasn’t for love, understanding and patience of my husband

Pavel and son Paul. I thank them from the bottom of my heart.

iv

Contents

1 Introduction 1
1.1 Introduction ................................ 1
1.2 Summary ................................. 6

2 Theorem on weak convergence to a diffusion process 8

2.1 A sufficient condition for weak compactness of a sequence of stochastic
processes .................................. 8

2.2 Theorem on weak convergence to a diffusion process .......... 9

3 Theorem on diffusion approximation for solution of perturbed dif-

ferential equation 14
3.1 Introduction ................................ 14
3.2 Assumptions ................................ 15
3.3 Results ................................... 17
3.4 Proofs ................................... 21
3.5 Example .................................. 37

Chapter 1

Introduction

1 . 1 Introduction

Randomly perturbed dynamical systems and differential equations were studied by
many authors: Krylov and Bogolubov, Gikhman (1950, 1951, 1964), Khasminskii
(1966), Papanicolaou and Varadhan (1973), Papanicolaou, Strook and Varadhan
(1977), Papanicolaou (1978), Krylov and Rozovskii (1979), Rozovskii (1990), Hop-
pensteadt, Salehi and Skorokhod (1995), Salehi and Skorokhod (1994, 1996) as well
as other authors. There are several types of problems that are considered for such
equations.

Averaging theorems state the convergence of the solution of the perturbed equa-
tion to the solution of the ”averaged” equation. The first averaging theorems were
proved by N.M. Krylov and N.N. Bogolubov in 19208 and 19303. They considered

the equation of the form

 

where function (1 depends on two times: fast (third argument) and slow (second

argument). If there exists the average of a in fast time

, 1 T _
Th—iEoT/O a(:r,t,7')d‘r —a(a:,t),

then the solution of the perturbed equation 3:6(t) converges to :Y:(t), which is the

solution of the averaged equation

 

Gikhman (1950, 1951) established a general averaging theorem for randomly per-
turbed equations. An averaging theorem for the randomly perturbed Volterra integral

equation

Mt) = W) + f K(s,t,y(§),x.(s>>ds

was proved by Hoppensteadt, Salehi and Skorokhod (1995). The kernel K depends
on a fast Markov process y with ergodic distribution p, which is assumed to satisfy
some mixing conditions. One of the results of the above mentioned article is the

convergence of 3176(t) to 5i:(t), the solution of the averaged equation

i(t) = (0(t) +/O K(s,t,§:(s))ds,

where

K(s,t,x)=/YK(s,t,y,;r)p(dy).

More precisely, it was proved that

P{lim sup Hut) — 53(t)|| = o} = 1.

6—)0 t<T

An averaging theorem for dynamical systems in Hilbert space was proved by Hop-
pensteadt, Salehi and Skorokhod (1996).

Hoppensteadt, Salehi and Skorokhod (1995) also present a result of another type:
a theorem on normal deviations. From the averaging theorem the difference between
the solution of the perturbed equation and the solution of the corresponding averaged
equations tends to zero. But appropriately normed, this difference (the deviation)
is asymptotically Gaussian. Namely, it is proved that the finite-dimensional distri-
butions of the process 6—1/2($6(t) — f:(t)) converge weakly as e —+ 0 to a Gaussian
process, which can be described as the solution of some stochastic linear integral
equation. A theorem on normal deviation for the difference equations was proved by
Hoppensteadt, Salehi and Skorokhod (1997).

The theorems on the approximation by diffusions deal with the asymptotic be-
havior of 2136(t/e). From the averaging theorems and theorems on normal deviations
we know that the solution of the perturbed equation is close to the solution of the
averaged equation and the deviation is approximately Gaussian with mean zero and
variance of order 6. When we consider 3:6(t) for large time, that is time of order
0(1/6), these deviations accumulate and 11:6(t/e) converges weakly as e —> 0 to a
diffusion process.

Theorems of such type were proved by Hoppensteadt, Salehi and Skorokhod (1995)

for perturbed Volterra equations, Salehi and Skorokhod (1994) for perturbed wave
equations, and Hoppensteadt, Salehi and Skorokhod (1997) for difference equations.
Salehi and Skorokhod (1996) consider a very general setup: operator differential

equations in a separable Hilbert space H:

dig)” = A(y(§))u.(t>. u.(0> = no (1.1)

 

and

d:;§‘>=At<g»u.m, u.(0)=uo, “it;

 

 

= ’00, (1.2)

where y(t) is an ergodic homogeneous Markov process in phase space Y with ergodic
distribution p satisfying some mixing conditions, and {A(y),y E Y} is a family of
closed linear operators with a common dense domain. The theory of differential
equations with operator—valued coefficients is contained in Krein (1982) and Kato
(1984).

Under the assumption that the operators A(y) commute and A*(y) = —A(y),
where A“ denotes the adjoint to A, the finite-dimensional distributions of u,(t/€)
coincide asymptotically with those of a Gaussian process {if (t). A similar result is
obtained for equation (1.2) under the assumption that all A(y) are symmetric non-
positive self-adjoint operators. Also, theorems on normal deviations are established
for both equations.

The method of proof is based on spectral decompositions for the operators A(y).

Since these operators commute, they can be represented as spectral integrals with

respect to the same resolution of identity. This method does not work if the operators
A(y) do not commute, which is the case considered in the present work.

We assume that A(y) = A0(y) + A1(y), where Ao(y) commute with
A = L, A(y)p(dy), and Al (y) are finite-dimensional with the same finite-dimensional
range for all y E Y. Since operators A(y), y E Y do not commute, we can not use
the method of Salehi and Skorokhod (1996).

A different approach using a sufficient condition for the convergence to a diffusion
process found in Skorokhod (1989). Such method is implemented in HOppensteadt,
Salehi and Skorokhod (1997) for the difference equations in Rd. It is based on martin-
gale characterization of multidimensional diffusion processes (see Strook and Varad-
han (1979)). Another important tool for proving the weak convergence is a sufficient
condition for the weak compactness of a sequence of the stochastic processes, which
can be found in Parthasarathy (1967) for the case of H -valued processes.

In the present work we generalize the method used in the paper of Hoppensteadt,
Salehi and Skorokhod (1997) to the case of infinite-dimensional Hilbert space using
the results of Daletskii (1967, 1983). The limiting diffusion process is an infinite-
dimensional diffusion process. Such processes arise in the theory of stochastic partial
differential equations. Stochastic differential equations have received a considerable
attention in the recent years. Here we mention just a few books and papers that
are devoted to them: Varadhan (1980), Rozovskii (1990), Protter (1990), Krylov and

Rozovskii (1979), Papanicolaou and Varadhan ((1973), Papanicolaou (1978).

1 .2 Summary

Chapter 2 contains the results that are necessary to prove the weak convergence to a
diffusion process in a separable Hilbert space H. Section 2.1 contains the theorem on
weak compactness of a sequence of H-valued stochastic processes, a condition that
ensures that the diffusion coefficients specify the transition probability of a Markov
process and a proposition on the martingale characterization of diffusion processes.
These results are used in section 2.2 to formulate and prove the theorem on weak
convergence to a diffusion process, which is the main result of Chapter 2.

In Chapter 3 we consider the differential equation (1.1) with operator-valued co-
efficients in a separable Hilbert space H. The coefficients are perturbed by an ergodic
jump Markov process y in a phase space Y with ergodic distribution p. We do not
assume that the coefficients commute, but assume that A(y) = A0(y) + A1(y), where
A0(y) commute with A = L, A(y)p(dy), and A1(y) are finite-dimensional with the
same finite-dimensional range for all y E Y. This assumption is crucial for the proof
of Lemma 3.7 (section 3.4). It ensures that all the integrals involving the resolution
of identity are well defined. We also suppose that A0(y) = a0(y)D, where no is a
real-valued function and D is a non-random Operator. This is a more restrictive as-
sumption than the one made in the paper by Salehi and Skorokhod (1996), where
a general family of closed commuting operators is considered. If A0(y) = 0 for all
y E Y, then we have a system of linear differential equations, which were studied by
Khasminskii (1966).

We think that it is possible to extend the results obtained in this dissertation

to the case when Ao(y) = ZZZlak0(y)Dk, n > 1, am k = 1,...,n are real-valued
functions and Dk, k = 1,. . . ,n are non-random operators, and also to the case of
general closed operators A0(y). These are the problems for future research.

Under the above assumptions on operators A0 and Al the commutativity condition
means that A0(y) commute with A1 for all y E Y. This condition is satisfied in many
applications when A1 = 0. An example of such kind is given in section 3.5, where we
consider a perturbed partial differential equation. When A1 aé 0, the commutativity
condition is not satisfied automatically. There are some applications of the results of
the thesis in this case as well (for example, to the systems of differential and partial
differential equations).

The main result is formulated in section 3.3. The proof is in section 3.4 and it is
based on the results from Chapter 2 and some special representations obtained using

the properties of Markov process y and spectral decomposition for the operator A.

Chapter 2

Theorem on weak convergence to a

diffusion process

2.1 A sufficient condition for weak compactness of

a sequence of stochastic processes

Let H be a separable Hilbert space and ("(t), t 6 12,. be a sequence of H-valued
stochastic processes. We say that 5,, converges weakly to a stochastic process 5 if the

finite dimensional distributions of {n converge weakly to those of the process 6, i.e.

"1320 Ef(€n(tl)a€n(t2)a - - °€n(tk)) : f(€(tl)v€(t2)v ' ° ° 6(a))

for all k 2 1, t1, t2, . . . tk E R+ and f : Hk —+ R that is bounded and continuous.

We say that the sequence {€,,(t), n = 1,2, . . .} is weakly compact if any subse-

quence {n.k, k 2 1} admits a further subsequence {nlk, k 2 1} such that 5",}: is weakly
convergent in the sense of the above definition.
To prove the theorem on weak convergence to the diffusion process we need a

theorem on weak compactness of a sequence of H -valued stochastic processes.

Theorem 2.1. Let the sequence of stochastic processes 5,, satisfy the conditions:
a) there exists a positive compact linear operator Q : H —-) H such that its range

contains 5,,(t) for all n 2 1 andt 6 R+ and

lim lim supsupP{||Q 5,,( (2‘)” > r} = 0, for all T > 0,

r—’°° n—+oo

1))

hm hm supsup sup P{||5,,() — 5n(t')|| > e} = 0
n—>oo t<T|t— t’|<h

forallc>0andT>0.

Then the sequence 5,, is weakly compact.

Proof of this theorem follows from the condition of compactness of measures in

Hilbert space (see Parthasarathy (1967), ch. V1, p. 151).

2.2 Theorem on weak convergence to a diffusion

process

We consider a Markov process 5 (t), t 6 12+ in H with transition probability

P(s,:r,t, B), :1: E H, 0 g s < t < 00, B E B(H). It is called a diffusion process if

there exist continuous functions a: R+ x H —> H and B: R+ x H —> L+(H), where
L+(H) is the space of all continuous non-negative linear operators from H to H, such

that

/g(:r')P(s,a:,t,da:’) — g(x) = [st/Lug(:r')P(s,:r,u,dx')du (2.1)

where g is a function from H to R that has bounded first and second derivatives,

0§s<tand

Lug(x) = (g'(a:),a(u,:r)) + $Tr(g"(x)B(u,x)) (2.2)

and TrB denotes the trace of the operator B. The operator L, is called the generator

of the Markov process.

Proposition 2.1. Let functions a and B satisfy the condition: for any r > 0 there

exists a constant l, for which

1/2

||a(t,a:) -— a(t,:1:’)|| + [Tr(B(t,:r) — B(t,:r'))2] S erI — 13'” (2.3)

if “2:” g r, ”513'“ S r, t S r. Then the transition probability is determined by functions

a and B through the formula:
P(s,:1:,t,A) = P{5~,,x(t) e A}, A e 8(H), a: e H, t 2 s 2 0,
where the process 55,2: is the solution of the stochastic differential equation

dam = a(t, 53,.(t))dt + B‘Wt,é.,x(t))dW(t) (2-4)

10

on the interval [3, oo) satisfying the initial condition

£3,x(3) : :13,

where W(t) is the generalized Wiener process in H for which E(VV(t), z) = 0,
E(W'(t), z)2 = tllzllz, z E H and BI/2 is a linear operator such that (Bl/2)"‘Bl/2 = B,
where B" denotes a conjugate to B. Under condition (2 3) the stochastic diflerential

equation (24) has a unique solution for any initial condition.

This proposition was proved by Daletskii (1967), Theorem 2.1, p. 33. Also see

Daletskii (1983).

Proposition 2.2. Let 5 (t), t 6 12+ be a measurable H -valued stochastic process and
let (1'), t E R+) be the filtration generated by 5. Iffor any function g : H -—> R with

bounded first and second derivatives

is a local martingale, where L, is defined by (2.2) and functions a and B satisfy the
condition of Proposition 2.1, then 5 admits a continuous modification 5, which is a

Markov random function with transition probability P(s, 3:, t, B).

This proposition is proved in Strook and Varadhan (1979) for Rd-valued stochastic
processes. The proof for the case of H —valued processes is the same.

Denote by C (2)(H ) the set of all functions from H to R with bounded first and
second derivatives.

11

Theorem 2.2. Let 5,,(t), t E 12+, n = 1, 2, . .. be a sequence of measurable H-valued
processes. Suppose that
1) the distributions of 5,,(0) converge weakly to some distribution m0(-) on B(H);

2) there exists a compact positive operator Q for which

lim lim supsup P{||Q‘15n(t)|| > r} = 0
r—ioo T

n-—>oo ts

for all T > 0;
3) there exists a subset D C C(2)(H) that is dense in C(2)(H) and the generator

of a diffusion process Lt with the coefficients satisfying the conditions of Proposition

2.1 for which

lim E(G(€n(t1),-~€n(tk)) 9(€n(t+h)) -g(€n(t)) -/t Lug(€n(U))dUJ) = 0

71400

forallkz 1, 091 g...tk <t<t+h, GeC(H"),geD.
Then 5,, converges weakly to a Markov random function 5 with the transition
probability P(s, x, t, B) that is determined by relation (2.1) and the distribution of

~

5(0) equals rn0(-).

Proof. It is easy to check that the sequence {5", n = 1,2, . . .} is weakly compact,
since the assumptions of Theorem 2.1 are satisfied.

Let {‘nl,’ ,l 2 1} be a subsequence for which the sequence {fun 1 Z 1} converges
weakly to some stochastic process 5. The distribution of 5(0) is m0 and 5 is a stochas-

tically continuous process, i.e. P{||5(t) — 5(8)“ > e} —> 0 as s ——> t. Therefore 5 has a

12

measurable modification, so we can assume that 5 is measurable.

It follows from assumption 3) of the theorem and stochastic continuity of 5 that

lim E(G(a<t1). . . aw) [gene + h» — gene» -— [m L.g(a(u))du}) = o.

71—)00

Therefore

E(g(é<t + h» — g(£<t>) — f L.g(£(u))du/a) = o,

and so the limit process satisfies the conditions of Proposition 2.2.
Since the sequence {5”, n 2 1} is weakly compact and all convergent subsequences

have the same limit, we conclude that 5,, —> 5 weakly as n ——> oo.

13

Chapter 3

Theorem on diffusion
approximation for solution of

perturbed differential equation

3. 1 Introduction

We consider operator differential equation in a separable Hilbert space H:

(3.1)
u,(0) = U0,

where {y(t), t Z 0} is an ergodic homogeneous Markov process in a measurable space
(Y, C) satisfying some mixing conditions and {A(y), y E Y} is a family of closed linear
operators with a common dense domain D, uo is a fixed element of D. We denote by

(~, -) the scalar product in H. Differential equations with operator-valued coefficients

14

are studied in Kato (1984) and Krein (1982).

Let p be the ergodic distribution of the process y. We assume that for all x E D

the integral f A(y)xp(dy) = Ax is defined, and we consider the averaged equation for

(3.1):

We will investigate the asymptotic behavior of u€(t/t) as 6 —+ 0.

3.2 Assumptions

1. Let {U(t),t 2 0} be a family of linear operators from H to H satisfying the
differential equation

10(15): amt), t > 0 (3 3)

where I is the identity operator.

The solution of equation (3.3) defines a semigroup of operators in H:

Assume that this group is unitary, i.e. for any f E D

(U'(t)f.U(t)f) = (f.f)-

Also suppose that {U (t), t E R} is weakly continuous.

II. Suppose that y(t), t Z 0 is a jump Markov process with transition probability
P(t, y, C), t Z 0, y E Y, C E C satisfying the relation
, 1
11m ?(P(t, y,C') — 10(y)) = H(y,C)

t—>0

and supy VarII(y, ) < 00.

III. SMC (Strong mixing condition). Set
R(t,y,B) = P(t,y,B) - p(B), t Z 0, y E Y, B E C.
Assume that
[000 |R(t,y,B)|dt < 00 for all y E Y,B E C.

Set

12(3), B) = A00 R(t,y,B)dt.

Under assumption I the group {U(t), t E R} admits the following representation

16

(see Dunford and Schwartz (1963), v.2, sec. XII.6.1, p. 1243, Stone’s Theorem):
Ur“) : eitS 2 / Bit/\dEA.
R

The resolution of identity E A and the symmetric operator S are determined uniquely
by the group {U(t), t E R}.
Set 141(3)) = A(y) — A, y e Y.

~ 4.

IV. Suppose that A(y) : A0(y) + A1(y), A0(y) = a0(y)D, where a0 is a function

from H to R for which

[/lao(y)|p(dy)<oo. Lflao(y)ao(y)3(y,dy)p(dy)<00.

D is a linear operator from H to H such that D“ = —D and A0(y)E,\ = EAA0(y) for
all y E Y.

Denote a0 = fY a0(y)p(dy).

V. Assume that A1(y) = iSl(y), and all S1(y), y E Y are symmetric and finite-
dimensional with the same range R of dimension ii. Let el, e2, . . . , en be an orthonor-

mal basis in R and Skj(y) = (S1(y)ek,e,-), k,j = 1,. . .,n.

3.3 Results
Set z.(t) = U(—t)u.(t).
Theorem 3.1. Suppose that conditions I—V are fulfilled. Then there exists a positive

17

compact operator Q such that the stochastic process Q2,(t) = Qz,(t/e) converges
weakly as c —> 0 to the diflusion process 2(t) with the generator L determined on

functions <I> : H —> R with bounded first and second derivatives by the relation

L<I>(Qz) = (6(ta 6262(2)) + §Tr<1>"(ez)eé(z)e (3.4)
where
62(2) = /Y [Y f/W} dial/lame) )dE.zR(y, dy )puy), (3.5)
3(2) 2 800(2) + 801(2) + 310(2) + 311(2), (3.6)
300(2) = 2 [Y [Y < Adz/)2 0 Ann/)2 > R(y.dy’)p(dy). (3-7)

Bale) = 2 f f // < Adz/)2 o dE1A1(y)dE,,z > R(y,dy')p(dy), (3.8)
Y Y {A=u}
B10(z) = 2/ / f/ < dEAA1(y')dE,,z o A0(y)z > R(y,dy’)p(dy), (3.9)
Y Y {A211}
B11(z) = 2/ / [f < dEAIA1(y')dE,,Iz o dEAA0(y)dE,,z >
Y Y {x+p—,\—p'=0}
>< R(y. dy’)p(dy)- (3-10)

Here < a o b > denotes the tensor product of vectors a, b E H, namely for any x E H

the following relation holds < a o b > x = (a, x)b.

Remark 3.1. For a function (I) : H —> R, its derivative <I>’ at point z is defined in

the following way: we consider <I>(z+tu), t E R, u E H as a function oft acting from

18

R to R. If the weak differential

D<I>(z, u) : gdflz + tu)
t=0

depends on u linearly, then D<I>(z,u) = (®’(z),u). Vector <1>'(z) is called the weak

derivative of (I) at point 2.
To define the second derivative consider
2

‘ <I>(z+t1u1+t2u2) , U1,U2€H, t1,t2€R.
dilatg ,l:0,,2:0

 

This expression {the second differential) is a bilinear function of ul, U2 6 H, and
it defines an operator <I>”(z) acting from H to H, this operator is called the second
derivative of function (I) at point 2.. Its boundedness means that the bilinear function

and the operator are bounded.

The third derivative is defined through

03
“(P(Z +L1U1+tQU2 + t3U3) 1' L’,(U1, U2, U3),
8t1at20t3 £120,t2:0,t3=0

ulau2iu3 E Hit1.t2,t3 E R-
The boundedness of the third derivative means that the trilinear form V defined

on H x H x H is bounded, i.e.

SUP ll’lfuliuzziusn = “V” < oo.
lluk||51.k=1,2,3

19

Remark 3.2. Under conditions IV and V, formulas (3.5) and (RU-(3.10) can be

rewritten as follows:

2) 2 D02 + f/ 2 [(,dE112 8j)AkjdE,\€k+

{A=#}k1;_1
dEA€k(ij2, dEuej) + SjkdEA€k(dEu2, 63)] ,

B00(z) = 2/Y/Yao(y)ao(y’)R(y, dy')p(dy) < D2 0 D2 >,

301(2. -)=2Z// d(Ez,e,-) <A,,,zodE,e,,>,

kj—l {3:11}

B10(2 — —2 Z // (,dEu2 Bj) < dEABk 0 ijZ >,

kj_1 id: fl}

811(2 — —2 Z //// Skj1m(dEu2,8j)(dE,/2,8m)X
{A’+p—A—u’=0}

k,j,.lm=1

< dExez O dEACk >,

where

20

i/ / &o(y)Skj(y’)R(y.dy’)p(dy)D,
Y Y

where 510(y) : a0(y) — ('10,

n
Sjk = E Sljkl-
(:1

Remark 3.3. Operator Q is chosen so that Q&(z) and and QB(z)Q satisfy the as-
sumptions of Proposition 2.1. Therefore Qa(z) and QB(2)Q determine the transition

probability of the process 2(t) as described in Proposition 2.1.

3.4 Proofs

The proof of Theorem 3.1 follows from Theorem 3.2 formulated below and the The-

orem 2.2 on weak convergence to a diffusion process.

Theorem 3.2. Let (I) : H a R have bounded <I>’, CI)”, <I>”’, D<I>’, DD<I>’, D<I>"D and

£(D<I>”(z)Dx,x), and let assumptions I—V be fulfilled. Then for any 0 3 t1 < t2

E(<1>(2.(t2)) — <I>(i.(t1))/f§.) = EU; L<1>(2.(T))dT/rg,) + 0(1),

where F,‘ is the o-algebra generated by {y(s/e), s _<_ t}, the operator L is given by
(3.4) with the coefiicients defined by formulas {3.5)~(3.10).

The proof of Theorem 3.2 is based on the following lemmas:

21

Lemma 3.1. The process z, is bounded, namely ||z,(t)|| = ||uO|| for all t > 0.

Proof. It is easy to see that z,(t) satisfies the following differential equation:

(3.11)

Note that B*(t,y) = —B(t,y) for all t 2 0, y E Y, where B" denotes the conjugate

to B, and so (B(t,y)z, z) = 0 for all t 2 0, y E Y, z E H. Therefore

(as), z.(t)) — (240), z.(0)) = 2 f (as), B<s,y(§))z.(s))ds = o.

C]

Let g be a measurable bounded function from H to R. Consider the linear oper-

ators H and R that act on a function g as follows:
H901) = fylgty') — 9(y)lH(y, dy')

R901) = [Y 9(y’)R(y. dy’).

22

where II(y,C) and R(y,C), 'y E Y, C E C were defined in assumptions II and III

respectively.

Lemma 3.2. Let g : Y —> R be measurable, bounded and satisfy

fY g(y (dy) - 0. Then IIRg— —

Proof. Consider the semigroup of operators {Tb t 2 0} generated by the transition

probability of the Markov process y(t):

119(3)) = / g(y')P<t,y.dy'),

then according to assumption II
1
H9 = 11111 t—(Ttg 9)

We calculate HRg under the condition that fr g(y )p(dy)— -— 0. Note that

fooo TiQU/ldt = Rg(y) and therefore

HRg- —— lim Oh(T),Rg— Rg)- — ling) h (/ ThT,g(y)dt - / T,g(y)dt) =
o 0

1 oo oo 1 h
1%h(/ T,g(y)dt-—/O T,g(y)dt)_ -— 1131—51) Ttg(y)dt= —9(y)

because T, ——> I as t —> 0, where I is the identity operator.

[:1

Lemma 3.3. The process (z,(t), y(f)) is a homogeneous Markov process in the phase
space H x Y with generator C, determined on functions f : H x Y —> R, which are

23

measurable and bounded with bounded partial derivative in the first argument, by the

       

relation:

G.(t)f(z,y)=11$%[E(f(z.(t+h),y( 1+1.» _ 2,, _,(:=,)_,(,,y,]=
: 1
(May),B(t.y)z)+;Hf(z.y) = (f: (z y) )+:/( m z y) (z ,’.y)]11(y,dy)

The proof of this lemma is the same as in the finite-dimensional case. The next

lemma is a generalization of Dynkin’s formula.

Lemma 3.4. Let f : R+ x H x Y —i R be measurable and bounded with bounded

partial derivatives in first and second arguments. Then for any 0 g t, < t2 < 00
t2 t1 6
E<f<t2. 26(t2).y(;)) — m1, z.(t.),y<;))/n) =

E(/[f:(s,z.(s),y(§)) + G.(s)f<s 24s ds / 7-1:)

Lemma 3.5. Let conditions I~V be fulfilled. Let (I) be a bounded measurable function

such that (IJ’ and D*<I>' are bounded. Then for any 0 3 t1 < t2 < 00

E(<I>(z,(t2)) — <I>(z€(t1))/.7-'fl) =

6E (I /} [(CI>”(z,(u))B(U,y(EeL-DZJU): B(u.y’)z.(u))+

+(<I>'(z,(u)), B(u, y')B(u, y(%))z,(u)) + (<I>'(z,(u)), B1(u, y')/lz — ABl(u, y')z,,(u)) x

24

u , 6
R(y(;),dy )du/r,,) + 0(6),
where B1(s, y) = U(—s).~’ll(y)U(s) and 0(6) does not depend on t1 and t2.

Proof. Using the fact that 26 satisfies the differential equation (3.11) we can write

<I>(z.(t2)) — <1>(z.(t1))= / 2 g(s, z.(s),y(§))ds

t1 6

where g(s, z, y) = (<I>’(z), B(s,y)z). Set f(s, z,y) = Rg(s,z, y), then since
fY g(s,z,y)p(dy) = 0 from Lemma 3.2 we have that IIf(s,z,y) = —g(s,z,y) (here

operator I1 acts on g as function of y). From Lemma 3.4 we have

E(/t2 g(s, z,(s),y(:))d8/.7:f,) = —e(f(t2,z,(t2),y(:)) — f(t1,z((t1),y(—)))+

t1

6

EU, [02(8. as), ye». B(s, y(§))z.(s)) + f:(s.z.(s), 11(3)] ds / F) =
E(/ l.» [(q>~(z.(u))B(u,y(§))z.(u).B(u,y')z.(u))+
+(<1>'(z.(u)), B(u, y')B(u, y(§))z.(u)) + (<1>'(z.(u)), B1(u, y’)/Iz — Asa, y')z,(u))] x

B(y(—E).dy')du/rt) +0(e)

since under the assumptions of the lemma function f is bounded.

Lemma 3.6. Suppose that conditions I—V are fulfilled. Then for a function

<1); H —> R such that <1>', <1)", <1>'" D*<I>’, D*D*<I>’, D*<I)”D and §;(D*<I>”(2)Dx,x) are

25

bounded, the following representation is valid:

EU; R(E,2,(T))dT/rg,) + 0(3)

where
K(r, z) = R’1(T, a) + Rg(T, z),
K1(Ta2) = [Y fy(<1>"(2)B(T.y)z.B(T.y’)Z)R(y.dy')p(di/),
1mm) = / jy (3(3), B(T. 338v,y)z)R<y,dy')p<dy).
Proof. Set

K(u, z,y) = fy[(<I>"(z)B(u, y)z, B(u,y')z)+
(‘1”(Z).B(U,y')3(uay)2) + (4"(2). 31W. y’)Az — fit-310‘, 3192)] B(y, 613/),
Kw, 2) = / Km. 3. z)p(dy).

Then we can rewrite the statement of the previous lemma as follows:

E(<I>(2,(t2))—<I>())/J-'§1)—E(/:2(K(s,(sz,y(st/ff)+0()

(

Denote P((s, z, y) = K(s, z, y) — K(s, 2). We need to prove that

£2
eE(/ t R’(s,z,(s),y(::-))ds/f§l) —+ 0 as c —+ 0.
El 6

26

Since fy H(s, z, y)p(dy) = 0 for all s 2 0, z E H, HRH = —K by Lemma 3.2 (the
operators H and R act on If as a function of its last argument, y).

Set f(s,z,y) = Rff(s,z,y). Recall that B(s,y) = a0(y)D + Bl(s,y), and B1 is
a bounded operator: Bl(s, y) = U(—s)/I1(y)U(s). Under assumptions of the lemma,
If has bounded partial derivatives in first and second arguments since 2:,E is bounded

(Lemma 3.1):

K;(s. 2. 32) = / [(3"(3)(3,(3, 3M — 31333, (2)2. 3(3, 3')z)+

(<I>"(z )3(3 92,) (3,(3 2M — AB.(s,i)z)+
(‘1”(Z).B(8.y')((31(8.iM-ABI(S.i/)Z)+(‘1"(Z).(31(S.y’)/5-431(8.y')Z)B(8.I))Z)+
(<I>’(z), B1(s, y'A/lz — 2AB1(s, y')Az + AAB((s, y')z) 12(3), dy'),

K;(s, y) = /Y 3*(3, y')B(s, y)z<I>'(z) + <I>"(z)B(s.y')B(s, y)+
[.13 _ AB,(3, y')]*<I>'(z) + mate _ AB-1(s, y')]z+
B(s.y’)*<1>"(z)B(s,y)z+B(s.y)*<1>"(z)B(s y)+{§((<1>"(z)8(s y) 3 B( (3 3):: )}|,_ z

From Lemma 3.4 we have

EU, K(,3 3,(3 y(—))—0d3/J-'f,) _

eE(/ [we z.(s).y(§)),B(s.y(§)) ( )) + f ( y(fw] / f)

27

So

Lemma 3.7. Under the assumptions of the previous lemma there exists the limit

~

T
limTaooi/ K(r,z)dr =K(z),
T 0

~ ~ ~

K(z) = K1(z) + K2(z), and
KHZ) :2 (A0(<I>"(z))z,z) + [/{A: }(A1(<I>”(z)dE,\)dE#z, z)+

f/{Azu} (A2(dEA<I>”(z))Z, dEflz)+

f/f/ (A3(dEAI¢"(Z)dEA)dE#Z,dEuIZ), (3.12)
{N+u-A-p’=0}

19(2) = [/3323} / fy(<1>(z),dEAA(y)A(y)dEyz)R(y,dy)p(dy) =
((D'(z),Doz) + //{ } Z [(dEpz,eJ-)(<I>'(z),.AikjdEAekH-
A=fl kgzl

((D'(z), dEAek)(C‘k,-z, dEuej) + 53.4%), dEAek)(dE,,z, ej) , (3.13)

where for a linear operator C from H to H

313(0) = h jy Marc/homo,dyipwy), (3.14)

28

343(0) ——- l3 f}an’rcfimaeu,dy'wy), (3.15)

343(0) -—- f [Y Alo'rczioomudy'wy), (3.16)
343(0) = fy/Y41(y’)*C41(y)R(y,dy')p(dy)- (3.17)

and

Do: [Y [Ydo(y')/io(y)R(y3dy’)p(dy)-

The operators A“, Ck], k,j = 1,. ..,n and coefiicients Skjlm, Sjk, k,j,l,m = 1,. . .n

are defined in Remark 3.2.

Remark 3.4. Assumption V ensures that the integrals in formulas (3.12) and (3.13}

are well- defined:

71

(A1(<I>"(z)dE,\)dE,,z, z) = Z (<I>"(z)dE,\ek, AkafldEyz, ej), (3.18)
kJ=1

A2(dE,V<I>"(z)z,dE#rz) = Z (dE#/Z,ej)(<b"(z)é'ka,dEXek), (3.19)
k,j=l

A3(dE,\,<I>”(z)dE,\)dE,,z,dEfllz)= Z Skflm(dEuz,ej)(dE,,z,em)x

k,j,l,m=1

(‘1’"(Z)dEAek, dExezl3 (3-20)

since for any vectors $1,172 6 H the expression (EA$1,.’E2) treated as a function of A,

has bounded variation on R.

29

Proof. Recall that

(,T z) —//(( (<I>”( (r y) z B(r y’) )R(y3dy’)p(dy)3

and B(r, y) = U(—r)xi(y)U(r). Using the spectral decomposition for (7(r) we can

K1(r,z) =LL(<I>"(z)/e_"’\dE,\}i(y)/eiT"dE"z,

/ (WWW) / e‘T“’dEirz)R(y,dy')p(dy> :

/ / [ff] eiT(,\’+l1—/\—p’)(A(y’)*dEAI@"(Z)dE,\/i(y)dE#Z,dEfltz).
Y Y

For an operator C set

= ff A(y')’Cfi(y)R(y3dy')p(dy)3

then

(,7' Z) “ff/f“ “W A #) (A(dE,\I(I)"(Z )dE),)dEMZ,dE“IZ).

A:

Under assumption IV zi(y) = Ao(y) + 341(31), and therefore A(C) can be written in

the form

“4(0) = A0(C) + A1(C) + Anzfcl + A3(C),

where A,(C), i = O, 1, 2, 3 are given by formulas (3.14)—(3.17) and satisfy the following

properties:

1) EAA0(C)Ep : A0(EACEp)a

30

2) EAA1(C) = A1(EAC)3
3) A2(C)Eu Z A2(CE#).

Using these properties we rewrite K1(r, 2) as follows:

(,7' z) =e'////T( “_AI‘L“ A ‘1)“ .Aoz(<I>"( ))dEAdEuz,dE,\rdE,,Iz)+

//// eiT(X+#-A—“’) (A1 (q)”(z)dE/\)dEuZ, dEAIdEuI Z) +
/./// eiTWfll-A—u’) (A2(dEA’(I)"(Z))dE,\dEpZ, dsz) +
//// €iT(A'+”—*-u’) (A3(dEz\’q)"(Z)dEA)dEuZ, dEpIZ) :

//(A0((I)I,(Z))dEAZ,dE/Vz)+

‘/'/V/6i T(—# A)(1(((D”(z)dEA)dEHZ,dEXZ)-i-

/// (WW—HI) (A2(dE,\"I>”(z))dE,\z, dsz) +

f/f/ eifiXW—W) (A3(dEx<I>”(z)dE,\)dE3z, dsz) 2

(AD (((PHZ )Z Z) +/e/ id” A) 1M1)" (ZldEA)dEuzaZ)-l—

ff WWW”(343(dEx<1>"(z))z, dE,,z)+

//// eiT(A'+“‘A"")(A3(dEN<I>”(z)dE,\)dEflz, dE#,z).

31

Now we average with respect to r, that is, compute

1 T -
Ef/O‘ h’1(T,Z)dT.

Note that

1 T .
Tlim _7: [0 e”(“_A)dr = 0 if ,u 7£ Aand it equals 1 otherwise.

Therefore

lim —/0T K( (,r z) =(A0(<I>"(z))z,z)+

T—+ooT

[fwd(A1(<I>”(z)dE3)dE,.z, 2) + df/{Azu}(A2(dE,\<I>”(z))z, dE,,z)+

ff/f («43(dEx<I>"(Z)dEA)dE,,z, 313,3).
{N+#->\—u’=0}

Formula (3.12) is proven.

Formula (3.13) is obtained similarly:

mm) = h fyo'e), B(n y’)B(T3y)Z)R(3/,dy’)p(dy) =

f, /Y f](<I>’(z),e-Wm(yawniwfl)my, 333,333) 2
f [W W ) >R(y3dy’)p(dy)+

/ f [/3110 “(‘1’A)Ao<y’>dEvil<y>dE3zm(3/,33333,),

/ / [/w A " NBA/A (:1 '>dEiAo(y)z)R(y,dy')p(dy)+

32

fyfyff eir(A—u)(<1)'(z),dEA/l1(y’).zll(y)dEpz)R(%(lg/)p(dy) 2

(drama) + f [Y // eI‘AA-A [ Z (Aug/idle),skj<y>(dE.z,e.)dE.e.)+

kJ=1

n

2' 2 (@'(z), Skj(y)(dE#/lo(y)z,ej)dEAek)+

k.j=1

n

i Z W(Z), Skj(y)(dEpzw ejldEA/il (31360] B(y, dy')p(dy) =

kJ=l

((1)’(Z):DOZ) + /Y /Y // eiTlA—M [i Z (dEuz,ej)(<I>'(z),30(y')Skj(y)dE,\ek)+

kJ=1

TI.

2' Z (Ao<y>z,e.><<1>'(z)(z), Stowe.)—

k.j—‘-1

Z Skj(y)(dEuZa €1>Szm(y')(ek, em)(<1>’(Z). dEAell] B(y, C1y’)/)(dy)-

k.j.l.m=l

Taking into account the definitions of 21k]. CH and SW", given in Remark 3.2 we have

Rg(7,z)=(q>'(z),00z)+/YL/faAA-W‘:

kj=l

[(dEpz, €j)((I)I(Z), fikjdEABk)+

We), dE.e.>(C“..z. due.) + slum e.><<1>'<z), diam] B(y, dy'>p<dy>.

Averaging with respect to T, we get formula (3.13).

Lemma 3.8. Suppose that the assumptions of Lemma 3.7 are fulfilled. Then

E(<I><2.(t2)) — wan/fig.) = E(/ 2 R’(2.(r)>dr/f-g.) + 0(1).

ii

33

where K() was introduced in the previous Lemma.

Proof. We need to show that

E(/t:2[R<-§,2.<s>>—fc<2.<s>]ds/ ;)

lim sup
c—->O

 

 

For an h > 0 consider

t+h 8 _ _ S
E5,z,y‘/t “K(E, 2£(S)) — K(Z~((S))) _ (K(Ev Z) — K(Z))]d8 (321)

From Lemma 3.7

t+h 8 ~
% f [12192) — K<z>1ds =
—/ [K(s, z) — K(z)]ds ——> 0 as 6 -> 0 for any h > O. (3.22)

Fix 3 > O and consider 1/2(z) = K(f, z) — K(z) as a function of z. The function 2/)

satisfies the assumptions of Lemma 3.5, and thus we have that

Bows» — man/f5) = E(<u>;(9>, 2.<s)> — (um), awn/e) = 0(e + h).
Therefore for some constant c1 > 0

P: {|(w;(0),2.(s) — mm > 6} s fS—‘(e + h).

;,Z,y

34

Thus

t+h s _ _ .9 ~
IE5.. / [um-5,24») — K(2.(s>>> — (K(—,z) — K(z)>]ds s

E

h(6 + fife + h)).

The last inequality together with formulas (3.21) and (3.22) imply that

8

1 t+h _ _ Cl
lim sup E5 / [K(—,E.(s)) — K(E.(s))]ds S 6 + ~6—h
t

e—+O c ,Z,yh 6

 

foranyh>0,6>0.

Set t]. =t1 +kh, k=0,1,2,...,n, then

E(/t[K(:§e(8)) — Mime/F1)

B(hg % [+1 [K(Eus» — K(2.(s)]ds/}'§})

for any 6 > 0, h > 0. The proof is completed by letting h —+ 0, then 6 —> 0.

lim sup
c—>O

 

 

S (t2 — t1)(5 +%1h)

lim sup
e—+0

 

 

E]

Lemma 3.8 implies the statement of Theorem 3.2 if we note that for any vectors

a, b E H and linear operator C from H to H

(Ca,b) =TrC<boa>,

and use formulas (3.12)—(3.17).

To complete the proof of Theorem 3.1 first we observe that the set of functions (I)

35

that satisfy the assumptions of Theorem 3.2 is dense in C (2)(H ) Therefore the first
part of condition 3) is satisfied.

Unfortunately, the drift EL and the diffusion operator 3 do not necessarily satisfy
the conditions of Proposition 2.1, and therefore do not necessarily define a diffusion
process. Therefore we consider the process Q§.(t), where Q is a compact positive

Operator, and we apply Theorem 3.2 to this process. Its generator
. 1 ~
L<I>(Qz) = (<I>'(Qz),Qa(z)) + §TTQ¢"(Q2)QB(2) =

(«<ta 6262(2)) + $Tr<I>"(Qz)QB<z)Q.

Note that a(z) is linear in 2, since it has the structure Flz, where F1 is a linear
operator. The diffusion operator B(z) is bilinear in z and has the structure < F22 o

z >, where F2 is a linear operator. Thus

Qa(z) = QFlz, TrQB(z)Q = TrQ < F22 o z > Q = (z,QF2Qz).

We choose Q so that QFI and QFgQ are bounded. Then Qa(z) and QB(2)Q satisfy

the condition of Proposition 2.1 and therefore define a diffusion process.

36

3.5 Example

Let H be L2(R3), that is the space of complex-valued functions f such that

[3 lf(:c)l2dsv < oo, :2: = ($1,252,133)-
R

Let A be the Laplace operator. We consider the equation

Bu.(t, :13)

8t : —iao(0(§))A + A1(0(§))u.(t,a‘), (323)

where 6(t) is a Markov process in phase space 9 that has ergodic distribution p and
satisfies conditions II and III of section 3.2, the function a0 satisfies condition IV, and
{A1(0), 0 E 8} is a family of finite-dimensional operators. For example, let A1(6) be

an integral operator with the kernel

1(xy,9=iZak(x)bkz/,

where ak, bk, k = 1, 2, . . . , n are real-valued functions and

u(,t 2:) —— —z':ak(:r )/3 bk( (y,0)u(t,y)dy.

Since the range of operator 141(6) is contained in the span of functions a1, . . . , an, this

operator is finite-dimensional. Hence the condition V of section 3.2 is fulfilled.

37

For convenience assume that

[gum/51510) =

Also assume that L, bk(y,6)p(d6) = 0 for all y E R3 and k = 1,2, . . .,n. The last
assumption ensures that condition IV of section 3.2 is satisfied: the Operator fl = —z'A
commutes with A009) 2 —ta0(0)A.

The operator S introduced in section 3.2 equals —A, it has the spectrum [0, 00).

Its Green function (resolvent kernel) is

 

1‘.
(a? 11): 4-7; lat-yl

13,3; 6 R3, :13 75 y, z ¢ [0, 00). The resolvent R. satisfies

(Ru,)v =//F(,:1:y) ())xdrrdy.
Ra Ra

Denote by E the resolution of identity of operator S. Using the expression for the
Green function and the fact that E ({a}) = 0 for all a E R, it can be shown that the

resolution of identity for the operator S satisfies

.1. 111mm Wm

Here C C [0, 00), u, v E C(2)(R3) and have bounded support.

 

Consider z.(t, a3) :2 e“‘Au.(t, :13). Let ej,j = 1, 2, . . . be an orthonormal base in L2.

38

The Fourier coefficients of z.(t, :13) can be computed as follows:

(Mikey) = 26.j(t) = (eitAu(t),€J-) =

1 _ __
foe it)A(alE.u.( ),e.-) =/ooe -AA / [Er—5 8‘“ (fix 3") 1.,(1 y)ej(:13)d)\d:cdy.
o R3 1234 l5” 9'

According to the Theorem 3.1, there exists a positive compact operator Q such

 

that the stochastic process Qz.(t/e) converges weakly to the diffusion process that is
determined by its generator. Let us compute the diffusion coefficients using formulas

given in Remark 3.2:

{1(2) = cAAz, c = — f6 [6(a0(6') — 1)(a0(0) — 1)R(0,d6')p(d9),

~

B(z) = Boo(z) 2 2c < A2 0 AZ >,

here

< AZ 0 AZ > f(:1:) = (A2, f)Az($) = Az(y) (y)dyAz(:13).

Ra
Remark 3.5. According to Remark 3.2 in the case of equation (3.23) 801(2) 2
810(2) 2 811(2) 2 0 and 51(2), 3(2) 2 Boo(z) do not depend on the operators A1(6),
since E({a}) = 0 for all a E R. So the limit process will be the same for all equations

of the form (3.23) as long as operators A(0) are finite-dimensional and the operator

A commutes with A.

Now let us choose an operator Q. It has to be chosen so that QAA is a bounded

operator. We can do it in the following way: we try to find a kernel K such that for

39

UELQ

QMI) = K(I. y)u(y)dy-

R3
Since Q has to be a compact operator, the kernel has to be in L2(R3 x R3), also it

has to be symmetric and nonnegative, and the following relation must hold:
IIQAAUII S Clllull (3-25)

for some constant c1. If the equation (3.23) is considered in a bounded region a C R3,
with the boundary 0a ( and this is usually the case for the equations of such form),
then we set K and AK together with their normal derivatives equal to 0 outside of a
and on its boundary. Then according to the Green formula for the Laplace operator

we obtain
QAAuo) = / K(z.y)AAu(y)dy =

6Au 8K
LAyK(:13,y)Au(y)dy +[3a(-57—1——K — Au%)ds —

[AyKCmi/MW/My: [AyAyK($.y)U(y)d1/,

where subscript y indicates that operator A acts on K as a function of y, ds is an
element of the boundary, and % denotes derivative in the normal direction to the
boundary. If AAK is bounded, then formula (3.25) is valid.

If equation (3.23) is considered in an unbounded region, then K and its derivatives

up to the third order have to decrease rapidly at infinity.

Remark 3.6. The example considered above can be generalized to the case ofa system

40

of partial differential equations. Let H be (L2(R3))’, that is the space of functions

u: R3 —> CT, u = (111,112, . ..u.)’ such that

|u(:13)|2d:13 < 00,
R3

 

here a: = (331,132,333), |u| = \/Iu1|2+ |u2|2+...|u,|2 is the Euclidian norm. We

consider a system of partial differential equations

Bu. (t,:13) , t _ " t
————gt : —’la§c0) (6(2))Aue,k(t, 317) + Z 2 (1.513(1) L3 bjk(y, 6(2))uc,j(ta y)dy

i=1

or in vector notation

8u6(t, .73)

. 0 t 1 t
at = —ia( )(6(;))Aue(t1$)+ A( )(6(E))u.(t,:13),

here a( — (11(0), . . .,aiol)’ is a vector-column. We assume that Is bjk(y,6)p(d0) = 0
for all j, k z 1, 2, . . .r and y E R3. Then condition IV of section 3.2 is satisfied.
Again, A(1)(6l) can be any finite-dimensional operators ( not necessarily integral) as
long as operator A commutes with A. Note that since the action ofA on (u1,...u,)
is coordinate-wise, the diffusion coeflicients of the limit process can be computed in

the same way as above, and they do not depend on operators A(1)((9).

41

Bibliography

[1] Daletskii, Yu.L., Infinite-dimensional elliptic operators and the corresponding
parabolic equations. Uspekhi Mat. Nauk 22 (1967) N4 (136) 3-54. English trans-
lation Russian Math. Surveys 22 (1967) 1-53.

[2] Daletskii, Yu.L., Stochastic differential geometry. Uspekhi Mat. Nauk 38 (1983)
87-111. English translation Russian Math. Surveys 38 (1983) 97-125.

[3] Dunford, N. and Schwartz, J .T., Linear Operators, Part II. Interscience Publish-
ers, New York, 1963.

[4] Gikhman, 1.1., To the theory of differential equations of stochastic processes. Ukr.
Math. J. 2 (1950), 37-63.

[5] Gikhman, I.I., To the theory of differential equations of stochastic processes. Ukr.
Math. J. 3 (1951), 317-339.

[6] Gikhman, I.I., Differential equations with random functions, In: winter school on
probability and statistics, Acad. Sci. Ukr., Kiev (1964), 41-86.

[7] Gikhman, 1.1. and Skorokhod, A.V., The Theory of Stochastic Processes. Vol. 1,
Springer-Verlag, New York, 1974.

[8] Hoppensteadt, F., Khasminskii, R. and Salehi, H., Asymptotic solutions of linear
partial differential equations of first order having random coefficients. Random
Oper. Stoch. Eqs., 2 (1994), 61-78.

[9] Hoppensteadt, F., Salehi, H. and Skorokhod, A.V., Randomly perturbed Volterra
integral equations and some applications. Stochast. Rep. 54 (1995), 89-125.

[10] Hoppensteadt, F ., Salehi, H. and Skorokhod, A.V., An averaging principle for
dynamical systems in Hilbert space with Markov random perturbations. Stochast.
Process. Appl. 61 (1996), 85-108.

[11] Hoppensteadt, F., Salehi, H. and Skorokhod, A.V., Discrete time semigroup
transformations with random perturbations. J. of Dynamics and Differ. Equations,
V0.9, 3 (1997), 463-505.

[12] Kato, T., Perturbation Theory for Linear Operators. Springer, New York, 1984.

42

[13] Khasminskii, R., On stochastic processes defined by differential equations with
small parameter. Theory Prob. Appl. 11 (1966), 211-228.

[14] Krein, S.G., Linear Equations in Banach Spaces. Birkhauser, Boston, MA, 1982.

[15] Krylov, N. and Rozovskii, B., On evolution stochastic equations. Itogi Nauki i
Techniki, VINITI (1979), 71-146.

[16] Parthasarathy, K. R., Probability measures on metric spaces. Academic press,
New York, 1967.

[17] Papanicolau, G., Strook, D.W. and Varadhan, S.R.S., Martingale approach to
some limit theorems. Duke Univ. Math. Ser. 3 (1977), Durham, NC.

[18] Papanicolaou, G., Asymptotic analysis of stochastic equations. MAA Studies 18,
Studies in Probabbility Theory (1978), 111-179.

[19] Papanicolau, G. and Varadhan, S.R.S., A limit theorem with strong mixing in
Banach space and two applications to stochastic differential equations. Comm.
Pure Appl. Math. 26 (1973), 497-524.

[20] Protter, P.H., Stochastic Integration and Differential Equations, Springer-Verlag,
New York, 1990.

[21] Rozovskii, B., Evolutionary Stochastic Systems, Nauka, Moscow 1985. English
translation Kluwer Academic, Boston, 1990.

[22] Salehi, H. and Skorokhod, A.V., On asymptotic behavior of solutions of the wave
equations perturbed by a fast Markov process. Ulam Quarterly, 2 (1994), 40-56.

[23] Salehi, H. and Skorokhod, A.V., On asymptotic behavior of oscillatory solutions
of operator differential equations perturbed by a fast Markov process, Probabilistic
Engineering Mechanics, 11 (1996), 251-255.

[24] Skorokhod, A.V., Asymptotic methods in the theory of stochastic differential
equations. Trans. Math. Monogr. 78 (1989), AMS.

[25] Strook, D.W. and Varadhan, S.R.S., Multidimensional Diffusion Processes,
Springer-Verlag, New York, 1979.

[26] Varadhan, S.R.S., Lectures on Diffusion Problems and Partial Differential Equa-
tions, Springer-Verlag, New York, 1980.

43

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