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2/05 chrTacmm.mms

SEMILINEAR STOCHASTIC DIFFERENTIAL EQUATIONS IN
HILBERT SPACES DRIVEN BY N ON-GAUSSIAN NOISE AND
THEIR ASYMPTOTIC PROPERTIES
By

Li Wang

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

2005

ABSTRACT

SEMILINEAR STOCHASTIC DIFFERENTIAL EQUATIONS IN HILBERT
SPACES DRIVEN BY NON-GAUSSIAN NOISE AND THEIR ASYMPTOTIC
PROPERTIES
By

Li Wang

A class of stochastic evolution equations with additive noise (compensated Poisson
random measures) in Hilbert spaces is considered. We first Show existence and unique-
ness of a mild solution to the stochastic equation with Lipschitz type coefficients. The
properties (homogeneity, Markov, and Feller) of the solution are studied. We then
study the stability and exponential ultimate boundedness properties of the solution
by using Lyapunov function technique. We also study the conditions for the exis-
tence and uniqueness of an invariant measure associated to the solution. At last, an

example is given to illustrate the theory.

Copyright by
LI WANG
2005

To my parents, my sisters, and Lan

iv

ACKNOWLEDGEMENTS

Foremost, I sincerely appreciate my advisor, Dr. V. Mandrekar, without whom
this dissertation would not have been possible. This dissertation comes from numer-
ous discussions in his office, from his keen insight and knowledge, from his guidance
to a fruitful research area, and his perseverance and support. His assistance in all
aspects of academic life was invaluable.

I would like to thank my guidance committee, Dr. Marianne Huebner, Dr. Den-
nis Gilliland, and Dr. Clifford Wcil for serving on my guidance committee. Your
help is highly appreciated. My special thanks go to Dr. Connie Page and Dr. Den-
nis Gilliland for training me as a statistical consultant. I would like to thank Dr.
Vincent Melfi, Dr. James Stapleton, Dr. Yimin Xiao, and Dr. Lijian Yang, who
taught me in many wonderful courses in statistics and probability.

Last, but not the least, I want to thank the department for offering me an as-
sistantship for five years. I want to thank all the professors and friends who ever

helped me during my stay at Michigan State University.

\f

Contents

1 Introduction 1
2 Preliminaries 4
2.1 Fréchet derivative ............................. 4
2.2 Gronwall’s inequality ........................... 5
2.3 Poisson random measure ......................... 5
2.4 Stochastic integral with respect to CPRM ............... 7
2.4.1 Stochastic integral for simple functions ............. 8
2.4.2 Stochastic integral for functions in H2 ............. 9

3 Existence and uniqueness of mild solutions to Semilinear SDE’s and
their properties 11
3.1 Mild and strong solutions of equation (3.1) ............... 12
3.2 Mild solutions of semilinear SDE’s with Lipschitz non-linearities . . . 13
3.3 Homogeneity, Markov, and Feller properties of the mild solution . . . 20
4 Approximating system 25
4.1 Sufficient conditions for a mild solution to be a strong solution . . . . 25
4.2 Approximation part ............................ 28
5 Stability properties of the mild solution 33
5.1 Ito formula ................................ 33
5.2 Exponential stability in the m.s.s ..................... 35
5.3 Stability in probability .......................... 46
5.4 Exponential ultimate boundedness in m.s.s. .............. 47
6 Invariant measures 56
6.1 Introduction ................................ 56
6.2 Existence and uniqueness of an invariant measure ........... 57
6.3 An example ................................ 68
6.4 Future research plans ........................... 69

BIBLIOGRAPHY 70

vi

Chapter 1

Introduction

The area of stochastic differential equations (SDE’S) in the infinite dimensional Hilbert
space with Gaussian noise was motivated by the study of stochastic partial differential
equations. Early contributions were made by Pardoux [35], Krylov and Rozovski [23],
Metivier [29], Viot [42], and Kallianpur et al. [19] motivated by Zakai’s [46] equa-
tion arising in filtering problems. These problems are described in Da Prato and
Zabczyk [6], who take the approach through semigroup methods as in Ichikawa [17].
The first study of the non-Gaussian noise case was done by Kallianpur and Xiong [20].
Their approach was to study SDE’s in duals of nuclear spaces where all bounded sets
are compact, but solutions turn out to be “generalized functions”. More general
equations are studied in Gawareski, Mandrekar, and Richard [10] extending the work
of Gikhman and Skorokhod [13].

In this thesis, we take the approach of Da Prato and Zabczyk and Ichikawa to
study the non-Gaussian case. We first, show that under the growth and Lipschitz

conditions, the uniqueness of solution in D([0. T]. H). We show that the solution is

homogeneous, Markovian and the transition semigroup is Feller. However, this is a
mild solution. In order to apply the recent ItO’s formula ( Rudiger and Ziglio [38] ), we
need to approximate this solution by strong solutions in C(O, T; L? (9,? , P)). Here
we adapt Ichikawa’s technique to generalize to our case Ichikawa’s work.

In the second part of the thesis, we study the asymptotic properties of the mild
solutions of these SPDE’s. We then study the Lyapunov function method first intro-
duced by Khasminski and Mandrekar [22] for the stability in the context of strong
solutions as in the works of Pardoux, Krylov and Rozovski. For semilinear equations
with Gaussian noise, this method was extended by Liu and Mandrekar [24]. They
also studied exponential ultimate boundedness of the strong as well as mild solutions
in Gaussian noise case. They showed that the ultimate boundedness can be used
to study the existence of invariant measure for strong solutions. In the second part
of the thesis, we study the extension of their work for non-Gaussian noise. We also
study existence and uniqueness of invariant measures for the case of mild solutions
in case the noise is non-Gaussian.

We begin in the next chapter by giving the definition of the Fréchet derivative and
stating Taylor’s theorem, followed by stochastic integral with respect to compensated
Poisson noise as presented in Rudiger [37]. Riidiger defines these stochastic integrals
for Banach space valued non-anticipative functions under some restriction. These
restrictions are removed in Mandrekar and Rudiger [28] where existence and unique-
ness of solutions of Banach space valued SDE’S with compensated Poisson noise are
studied.

In Chapter 3, we study the existence and uniqueness of a mild solution to the

semilinear SDE’s under the growth and Lipschitz type conditions on the nonlinearities.
We then study the homogeneity, Markov, and Feller preperties of the mild solution.
In Chapter 4, we study the approximating system of the semilinear SDE’s. We prove
that there exists a unique strong solution to the approximating system, and the
strong solution converges to the mild solution in C(0,T; L51 (9,? , P)). In Chapter
5, we study the stability and exponential ultimate boundedness in the m.s.s. of the
mild solution by using the Lyapunov function technique. We use Ito formula and the
approximating systems as tools. We first prove that the existence of the Lyapunov
function is sufficient for the stability and exponential ultimate boundedness in the
m.s.s. of the mild solution. Conversely, we construct the Lyapunov function for the
linear case, and construct the Lyapunov function for the nonlinear case by using the
first order approximation of the coefficients. In Chapter 6, we first give the conditions
for the existence and uniqueness of an invariant measure associated to the solution.

Finally, we give an example to illustrate our theory.

Chapter 2

Preliminaries

In this chapter we recall some definitions and theorems that we will use in our theory.

2. 1 Héchet derivative

The results below are adapted from Schwartz’s work [39].
Let H be a real separable Hilbert space, we use (, ) and I] ' [In to represent the
inner product and norm in H. L(H) denotes the set of bounded linear operators from

H to H. Assume f: H -—> R is a map and :r,y E H.
Definition 2.1.1. We say that f is first order Fréchet dz'flerentiable at 2:, if there
exists an f’(.r) E H, such that

[(-F +11) = [(I) + (f’(.::), y) + "(III/H11)-
We say that f is second order Fréchet differentiable at :17, if there exists an f”(17) E
L(H), such that

f’tr + y) = f’(-T) + f”(:17)y + (’(IIIIIIIHl-

Definition 2.1.2. A function f : H —) IR is said to be in class C2 on H, written
f E C2(H), iff f’(:1:) and f”(:1:) exists at every point of H and the maps :1: —+ f’(.r)

and a: -> f ” (:r) are continuous.

Theorem 2.1.1 (Schwartz [39]). Suppose that f E C2(H). Then

1

f(r + y) = f(r) + was), y> + j <1 — t><r<x + any, y> dt.

0
Corollary 2.1.1 (Schwartz [39]). Suppose that f E C2(H). Then there exists a

bounded bilinear function R2 from H to R such that

ffzr + y) = ft”) + (f’fT), .11) + It32(9)-

2.2 Gronwall’s inequality

Theorem 2.2.1 (Evans [9]).1ffort0 _<_ t S t1,¢(t) 2 0 and tl’(t) 2 0 are continuous

functions such that the inequality

t
¢(t) s K + L/t ¢(s)¢(s)ds

holds on to S t _<_ t1, with K and L positive constants, then on to S t 3 t1,

(PU) S Kexp (L [t w(s)ds).

.to

2.3 Poisson random measure

Let (0.1:,{55 LEO, P) be a filtered probability space satisfying the “usual hypothe-

ses” :

1. ft contains all null sets of .77, for all t E [0, 00),

zfl=fimmwfi=f]aamm6pm.
u>t

Definition 2.3.1. Let (X, X) be a measurable space. A map: N : Q x X ——> R is

called a random measure if
1. N(w, ) is a measure on (X, X) for each u) E Q,
2. N ( -, B) is a random variable for each B E X.

Definition 2.3.2. A random measure N is called independently scattered if for any
disjoint 81,...,B,, E X, the random variables N(-,Bl), ,N(',Bn) are indepen-

dent.

Let (E, 8) be a measurable space (E is a complete separable metric space), and let
the map: N : Q x (8 x B(lR+)) —+ IR be a random measure, with X = E x IR+ and

X=s®3mn.

Definition 2.3.3. The random measure N is adapted if N (-, B) is ft—measurable for
B C E x [0, t]. N is o-finitc if there exists a sequence En increasing to E such that

E]N(-,E,, x [0,t])] < 00 for each n E N and 0 < t < 00.

Definition 2.3.4. The random measure N is called a martingale random measure if
for fixed A 6 I‘N :2 {/1 E 8 : E|N(/I x [0,I])| < 00, V 0 < t < 00}, the stochastic

process N(A x [0, t]) is martingale adapted to {.71 it20-

Let A denote the collection of all E-adapted processes whose sample paths are of

finite variations on any finite intervals.

Definition 2.3.5. A o—finite adapted random measure N is said to be in the class
(QL) if there exists a unique o-finite predictable random measure N such that N :=
N — N is a martingale random measure and for any A 6 FN, N (A x [0, t]) E A and

is continuous in t. The random measure N is called the compensator of N.

Definition 2.3.6. Let u be a o-finite measure on B(E x IR+). The Poisson random

measure is a random measure N : (I x B(E x IR+) —> IR, such that:

1. It is a independently scattered nonnegative integer-valued adapted random mea-

sure.

2. If for any B 6 EU? x IR+) such that u(B) < 00, N(-, B) is a Poisson random

variable with mean v(B).

Definition 2.3.7. Let 5 be a o-finite measure on (E,8)( with fi({0}) = 0 and
5(8) < 00, if B E B(E) and 0 Q“ B, B represents the closure of B). If we suppose

v(A x [0, t]) = l3(A)t for any A E 8, then B is called the characteristic measure.

It is clear that any Poisson random measure N is in class (QL) with the compen-

sator N(A x [0, t]) = I3(A)t for any A E 8. And
Mia/4 x [0.tl) .= N(w,A x [0,1]) — N(w,A x [0,t])

is called compensated Poisson random measure (CPRM, for short).

2.4 Stochastic integral with respect to CPRM

In this chapter, we will introduce the stochastic integral with respect to CPRM fol-

lowing Riidiger [37].

Let Ft 2: B((E\{ 0 }) x lR+) ®ft be the product o-algebra generated by the semi-
ring B((E\{ 0 }) x IR+) x T; of the product sets B x F, B E B((E\{ 0 }) x IR+), F E .7}.
(A ring is a non-empty class of sets which is closed under the formation of unions
and differences)

As before let H be a real separable Hilbert space, and let (-, ) and I] ° ”11 represent
the inner product and norm on H separately.

Let T > 0 and let
MT(E/H) :2
{f: (E\{ 0 }) x IR+ x D -—> H, such that f is FT/B(H)-measurable

and [(35, t,w) is ft-adapted V1: 6 E\{0} Vt E (0,T]}

2.4.1 Stochastic integral for simple functions

Definition 2.4.1. A function f belongs to the set 2(E / H ) of simple functions, if

f E MT(E/H), T > 0 and there exist it E N, m E N, such that

3
p—l

m

“M = lAk,(zl' )IFk,(I “)(tht-HIUMH
1 (=1

3-
II

where AU 6 B(E) (0 ¢ A17), It 6 (0,T], Ik < Ik+1, F“ E ftk, a“ E II. For all

k=1,...,n-—1fixed, Ak,l1 kaJlnAkJ2 XFk,12=¢,if117él2.

Definition 2.4.2. For the simple function f E 2(E/II), we define the stochastic

integral with respect to CPRM by

f/fxthdtdt =

for all A e B(E\{0}), T > 0.

n— 1

ZakHZIFklMOIV(/1kln A X (tk,tk+1]fl (0,T])(UJ)
k=1 (:1

2.4.2 Stochastic integral for functions in H2

To define the stochastic integral for more general functions than simple functions, we

give some definitions first.

Definition 2.4.3. Let L51 ((2, f, P) be the space of H —valued random variables, such
that EHYHE, = f]]Y||§, (1P < 00. We denote by I] - ”2 the norm given by ]]Y]]2 =

(E]|Y|]%)1/2. Given (Yn),,eN, Y E L§I(Q,.7:, P), we write lim2 Y = Y if lim [lYn —

n—ooo '1

v”2 = 0.

Definition 2.4.4. Let f : (E\{0}) x lR+ x D —-> H be given. A sequence {fn },,6N
of FT/B(H)-measurable functions is L2-approrimating f on A x (0,T] x Q w.r.t.

5 53> Leb 69 P, if fn is 5 ® Leb 8 P-a.s. converging to I, when n —* 00, and

T
hm / [A Ellfn(:r.t.w)-f(:c,t,w)ll313(dx)dt=0;

n—‘OO

i.e., ||f,, — fl] converges to zero in L2(A x (0,T] x Q, i3 63) Leb ® P), when n —> 00.

Next, we define the class of functions on which we will define, the stochastic inte-

gral.

H2 2:
{f(:c. t.w) : (E\{ 0 }) x R. x n —+ H, such that f is FT/B(H)-measurable
and f(.1:, his) is ft-measurable for Va: 6 E\{ 0} and VI 6 (0, T], also

T
2 (E C .
E f / llf(I,taw)Hufi(d )lt<oo}

Remark. Observe that if f 6 H2, then for each t 6 [0,7], 1[0,t1f E Hg. When it’s

clear from the context, we write elements of 1/2 as f (1:, I).

Let f 6 H2. Then there exists simple functions { fk} 6 H2, such that

t
E// |]fk(:r,s,w) —f(:r,s,w)]|§1,8(dx)ds —) 0 as k —> oo,
0 E
And the simple functions { ft} satisfy, for each t E [0, T]
t
EHMW) - L(lelliz = E/O/Ellfk(r,8) -fj(I,S)lIiI M6133) d8
—+ 0, as k,j —> oo.

( See Rudiger [37] )

Now for f E [‘12, we define the stochastic integral with respect to CPRM as,

It(f) = /0 /Ef(r,s)N(da:ds) =limi._,oo It(fk).

The following results are known about the stochastic integral with respect to

CPRM.
Theorem 2.4.1 (Riidiger [37]). Let It(f) be defined as above. Then we have
1. The sample paths of L(f) = fng f(.r,s) N(dx ds) are cddlag,

2. I,( f ) is a mean 0 martingale with respect to .7}.

10

Chapter 3

Existence and uniqueness of mild
solutions to Semilinear SDE’s and

their properties

In this chapter, we will consider the following semilinear stochastic differential equa-

tion ( SDE, for short) on [0,T].

dZ(t) = (AZ(t) + F(Z(t)))dt +/ B(v,Z(t))N(dvdt), 2(0) = «,9 e H. (3.1)

Here A is the generator of a CO-semigroup S (I) on H, satisfying ]]S(t)]] g e‘“, a 6 IR.
N is a CPRM on E X IR+. F and B are, in general, nonlinear mappings, F : H —> H,
B : E X H —+ H.

Remark. Partial differential equation can be written as Hilbert space valued linear
equafion.

In this chapter, we will prove the existence and uniqueness of the mild solution to

11

the system under some conditions on the coefficients, and we will also prove that the

solution has homogeneity, Markov and Feller properties.

3.1 Mild and strong solutions of equation (3.1)

Let T = [0, T], we first introduce two types of solutions.

Definition 3.1.1. A stochastic process Z (t), t E 'II‘, is a mild solution of (3.1) if
(i) Z(t) is adapted to .731,
(ii) Z(t) is measurable and If E|]Z(t)||§, dt < 00,

(iii) Z(I) = sup + f,,‘ so — s)F(Z(.s-)) ds + [ng 3(t — 93(7), 2(3)) Minds) for all

t E T, w.p. 1.
Definition 3.1.2. A stochastic process Z (t), t 6 T, is a strong solution of (3.1) if
(i) Z(t) is adapted to E,
(ii) Z(t) is cadlag in t, w.p. 1,
(iii) Z(t) E ’D(A) almost everywhere on T x Q, and fOT ||AZ(I)|]H (it < oo, w.p. 1,

(iv) Z(t) = cp + fot AZ(s)ds + fot F(Z(s))ds + frills B(v,Z(s)) N(dvds) for all t E

T, w.p. 1.

12

3.2 Mild solutions of semilinear SDE’s with Lip-

schitz non-linearities

We impose the following assumptions on the coefficients of equation (3.1)
(A1) F and B are jointly measurable,

(A2) There exists a constant I, such that Va: 6 H,
Haunt + [E I|B('v,r)llii fi(dv) 5 «1+ lent),

(A3) For all 1’, y E H, there exists a constant k, such that

 

 

FOE) - F(3/)||it + [E H3051?) - 13(v»y)|Ii1I3(d'v) S klla: - yllii-

We prove the existence and uniqueness of a mild solution to stochastic equation
(3.1) under the above conditions. We follow the ideas from the work of Gikhman and
Skorokhod [13], and adapt them to our case.

Observe that for d 6 H2, foth S(t — s)d>(v. s) N(dvds) exists because

t
f / use — salesman. < oo.
0 E
The following lemma comes from Ichikawa [16].

Lemma 3.2.1. Let N be Poisson random measure and S (I) be a pseudo-contraction

semigroup. Assume (f) 6 H2. Ifr is a stopping time, then

2

TAT
E sup SbIE/ / lletv,t)llin3(dv)dt-
H 0 E

OSth/‘vr

 

 

/0t/E S(t — s)c_b(v, s) N(dvds)

 

 

 

S(l)]] S e‘”.

 

The constant b1 depends only on T and (r as in the bound

13

Proof. Let M, = fost q5(v,r) N(dvdr) and y¢= foS( (t— s) d)M. From Ichikawa [16],

we know if a g 0, then

E(Sl<_1pllytl]H)<3+\/_)2 <M>

E(sup fo/Efl ()t—sqb ¢)(,v s) )N(dvds)
t<r

3(3 + «EVE/01A; ]]¢(v, .5)”; [3((112) ds.

Hence

 

 

Z)

 

 

If a > 0, we use the results in Ichikawa [16] and get

2
E (sup Ilytllfi) S e207°° (3 + v10) E(M),.
tSr

Ifere

ess supr if it exists

oo otherwise.

Now if we use T /\ r in place of r and notice that (T /\ r)oo S T and let bl =

e2luIT(3 + \/1_0)2, we get

[0/S( ()t—sgb o(i,s)N(duds)

 

 

 

 

TAT
:) _<_ blE/ /|]gb(s,u)]|§,,3(dv)ds.
0 E

C]

811p
0<t<TAr

Now for an adapted process €() 6 D([0,T],H), satisfying Esupogng |]{(s)|]%,

< 00, we let

wimp/0t (t—s)F £(s))ds+//S( (t—s)B (a “(wields

Remark. The above integral exists under (Al) and (A2).

14

Lemma 3.2.2. IfF(:c) and B(v, :r) satisfy (A1) and (A2), S(t) is a pseudo-contraction

semigroup, then for a stopping time T,

E( sup III<s,é<s>>II%.)Sb2(+ +/E sup Il€(u)||%ds)

OSsstAr O<u<sAT

with the constant b2 depending on a, T and l.

Proof. We note that

2

 

 

SUP ||1(S.€(S))llh S? SUP (

OSSSMT OSsStA‘r

 

[3 S(s — u)F(§(u)) du
0 H

+/0/ES( ()s—u B(,v€(u )) N(dvdu) H).

There exists a bound for the expectation of the first term, using “S(t)” S e‘", the

 

 

 

 

 

Schwartz inequality and (A2),

 

 

 

 

s 2
E sup /OS(s—u)F({(u))du
O<s<tAT H
2
<E sup {/IlSHs—uF( ({(u))||Hdu}
O<s<tAT

       

2
<E (I It
_ (fome mnlmu)
S(z2atE(t /\ T) [0 ||F(§(u.))||§i (1n
Seza‘tIE / T<1+ nau>nt>du

0

t
se2°‘tl(t+/ E sup ll€(u )llyds)
0

UsuSsA‘r

From Lemma 3.2.1 and (A2), for the second term,

/0/E( S(s ‘ “)B(v=€(u)) N(dv du)
SblE /M [BMW 3))HH 5((1’v)ds

15

2
E sup

0<s<tAT

 

 

 

 

H

:13le / T<1+ Ilasmi’nds

ss.z(t+/ E sup |l€(u)||31ds)-

OgugsAr
Let b2 2 2e2lalTTl + QezlalT(3 + my. We complete the proof by combining the

inequalities for both terms. El

Lemma 3.2.3. Let conditions (AI) and (A3) be satisfied and S(t) be a pseudo-

contraction semigroup. Then

Eos<upll1(8.€1(8)) — I<se<s>>nt 3 b3 / E0s<ups l|€1(u) — smut ds

with the constant b3 depending on oz, T and k.

Proof. We begin with the following estimate.

E sup ll1(8,€1(8)) — “342(5)”?!

 

 

 

 

 

 

 

 

 

 

 

 

s2E0sgggt( / S(s — wales» — momma .1
+ [0 >/ES(s — 1.1.)[B(1I,€1(u)) — B('U,§2(u))] N(dvdu) H).
As before,
E sup f S(s — u><E(:.<u>> - new)» du
OSsSt O 11
@0231] use — u)“ IIF(€1(u)) — woman” a)?

seam sup / IIF(€.(II))—F(€2(II))II%I1II
0

Ogsgt

skemtE sup / Ile.(u)—a<u)utdu
0

Ogsgt

t
3136th j E sup new) —s.<u>ui.ds

Ogugs

16

And,

2
E sup

OSsgt

beE/[EIIB(U,€1(S))-B(v,€2(8))ll§fi(dv)ds

 

 

 

 

if}; S(s — u)[B(v.€1(u)) — B(v,€2(u))] N(dv du)

If

t
3ka f E sup I|€1(u)—€2(u)lltds-

Ogugs
Let 1);, = 2e2l"lTT k + 2ke2lalT(3 + \/1_0)2. We complete the proof by combining the

inequalities for both terms. [:1

Remark. Lemma 3.2.2 and Lemma 3.2.3 follow from the arguments similar to the
arguments for Brownian motion case by Gawarecki et a1. [10].

N ow we prove the existence and uniqueness of the mild solution.

Theorem 3.2.1. Let the coefficients F and B satisfy conditions (A1), (A2) and (A3),
assume that S (t) is a pseudo-contraction semigroup. Then the stochastic equation

(3.1) has a unique mild solution Z(t) satisfying
t t ~
Z(t) = S(t)cp +/ S(t — s)F(Z(s))ds +/ / S(t — s)B(v, 2(3)) N(dv ds)
0 o E
in the space
H2 2: {€() 6 D([O,T], H), such that E sup “6(5)“; < oo}.
ogng

Proof. We follow Picard’s method. Let I be defined as before. By Lemma 3.2.2,
I : H2 —-> H2. The solution can be approximated by the sequence Z0(t) = (,0, - -- ,
Zn+l(t) = 1(t7Zn(t)). TL : 0,1,° ' ' - LQL V710) : ESUPogsgt l|Z71+1(S) — 271(8)“?!

Then V0(t) = ESUPogsgt ||Z1(s) — Zg(s)||§, S VO(T) E V0, and using Lemma 3.2.3, we

17

obtain

V10) = E SUP “22(3) - 21(8)”?1

OSsgt

= E sup ||I(s, 21(8)) — 1(83Z0(8))ll§{

OSsSt

t
3 b3 / E sup HZ1(u)— 2001.)”?st g b3V0t.
0

OSuSs

V0(b3t)n

' . Next, similar to the proof of
n.

t

By induction, Vn(t) _<_ b3/ Vn_1(s)ds S
0

Gikhman and Skorokhod [11], we show that SUPogth ||Zn(t) — Z(t)|lH —> 0, as. for

some Z 6 H2. If we let 5,, = (V0(b3T)”/n!)1/3, then, using Chebyshev’s inequality, we

arrive at
V b T " V b T " 2/3
Pt sup nz...< >— Emu” > a.) s (—°(—3,)—)/(i(3,-)—) = (3.2)
0<¢<r n. n.
Because 2:: 5,, < 00, the series 2‘” _lsup0<,<T ||Zn+1(t )— Zn(t)||H converges as,

showing that Zn(-) converges to some Z() a.s.. By (3.2)

2H sup Hz...1<>— awn” >s.> < oo.

0<¢<T
By Borel-Cantelli lemma, we have SUPogth ||Zn+1(t) — Zn(t)||H < 5,, as. So we have
”Z(t) — Zn(t )HH —* 0, a. 3.. Because Zn(t) has the cadlag property, the sample paths
of Z (t) are cadlag.

Moreover,

E sup ”Z(t) — Z110)”;

093T

=E lim sup HZn+m( )— 271“)”th

 

 

m_’°°0<t<T
n+4n—1 2
=E lim SUP Z (anal—21(0)
mfiooostST k=n H

 

 

18

n+nr4

SE49“ 2 sup _IIIzI..II>— zIIIIIIHY

n+n1—l

=nliggoE( Z sup IIZI..II >— ZIIIIIIHIs-if

O<t_
=n

S g £7021ng ||Zk+1(t) — Zk(t)llf1k2(;k—2)’

and hence the second series converges. The first series is bounded by observing

V (bT’ck2
EMT T)k2<Z—0—3—)——+0,.asn—>oo

k=n k=n

To justify that Z (t) is a mild solution to equation (3.1), we note that as. F (Zn(s)) —>

F (Z (3)) uniformly in 8. Therefore

1) S(t — s)F(Zn(s)) ds —>/0 S(t — s)F(Z(s)) ds a.s..

Using the fact proved above, then

E sup IIZIII — anIIIIiI —+ 0.

05th

Thus we obtain

Esu — .s)[B(II, 2(3)) — B(II, z,(..-))] N(IIII ds)

 

 

       

it

ssIE / / ||B( s ZIs B(v 2 Is ))l|?sfl(dv)ds

SblkTE sup “Z(t) — Zn(t)||§, —> 0, as n ——> oo.
()5th

This solution is unique. lf Z (t), Z ’(t) are two solutions to equation (3.1), then we
define V(t) = ESUPogsgi ||Z(s) — Z’(s)||2H. By Lemma 3.2.3,

t b t n
V(t)Sb3/ V(8)dsS---SE sup IIZIsI—Z'IsIIIt( 3,.) —»o,
0 .

OSSST

 

as n -—> 00, giving V(t) = 0. El

19

Remark. We know that there exists for each T < 00 an unique mild solution of

(3.1). We can define Z(t) on [0, +00), such that for any T, E511Pogsgrllz(5)llH < 00.

3.3 Homogeneity, Markov, and Feller properties of

the mild solution

Notation. From now on, we will use Z‘p(t) instead of Z (t) to represent the mild

solution of (3.1) to emphasize that the the solution depends on the initial value (,0.

Lemma 3.3.1. The mild solution of (3.1) Z‘I’(t) is continuous in the initial value «,0

(w.r.t. the strong topology on H).

Proof. Let cpl and 4,22 6 H be the initial values. Suppose that the two mild solutions

are 2991(1) and 2220). Then we have
221(I) = S(t)s21+./OtS(I— s)F(Z¢1(s))ds + [A S(t — SIBIII, 221(3)) N(dII ds),
and
z22III = EIII Is. + f EII — sIFIz22Is-II Is + [if SII — sIEIII, z22IsII NIIIIIIIII.
Thus
Z2III I— We )
=SIII I « — s.) + f EII — sIEIz22Is II IIs

/ f(t—sfli’vZf1())N(dvds)

_. (f S(t — S)F(Z‘102(S))Il.s + [O/ES(I— S)B((IZK102( )) N((ll- (15.))

0

2O

From Lemma 3.2.3 we have

E sup IlZ901(8) — 2802(5)”;

OSsSt

SE sup(2||3(8)(991- 902)||31 + 2||I(s, Z9048» - 1(8I Z‘WSDHE)

055st

t
s2s22IIs. — szIIt + 2s f E sup IIz22IIII - Z22IIIIIIII Is.

0 OSuSs

By Gronwall’s inequality we have

E sup “221 IsI — Z22IsIIIII s 2e22tIIsl — s2IIte2b2‘ = 2422 + 2b3lt|l901 — Issui-

Ogsgt

So the solution is continuous in the initial value. Cl

Theorem 3.3.1. The mild solution of (3.1) is homogeneous in t and it has the Markov

property.
Proof. Fix 5. Let us denote by (Zs"”(t)),2s the solution of

ZS"'°(dt) = (AZ3"”(t) + F(Zs"'°(t))) dt + /E B(v, ZS“p(t)) N(dv dt), Zs"'°(s) = go.

Following Theorem 3.2.1, it can be checked that such a solution exists and is unique
up to stochastic equivalence. Let us remark that the compensated Poisson random
measure is translation invariant in time; i.e., if h > O, £(N(v,s + h) — N(v,s)) 2

ma, II).

It follows that
s+h
Z‘W(s + h) =S(h)',.c +/ S(s + h — u)F(ZS"p(u)) du
s+h 5 ~
+/ /S(s + h —- u)B('LI, ZS“'°(u)) N(dvdu)
s E

h
=S(h),c + /O S(h — II)F(22I2I3 + II» In

21

+/h /ES(h — u)B( v, ZS"(s+u))l§(dUdu)2 (3'3)

Here N(v, u): N(v, s + u)— N(u,s).

From Theorem 3.2.1, we have
h
ZO"p(h) =S(h)<,o+/0 S(h—u)F(ZO"'°(u))du
h
— u 0"pu ~ 1) u. .
+/0 [15801 u)B( ,Z ())N(d d) (34)

As the solution of (3.3) and (3.4) are unique up to stochastic equivalent and N (duds)
andN(dvds) are equally distributed, it follows that {Z02‘P(h) lth and
{ Z32"°(s + h) }h20 are stochastic equivalent.

We have proved that {Z020 (t )}I>0 is cadlag. Let T > 0. We denote by Q‘p
the distribution induced by {Z 02¢(t) }te[0.T] on the Skorokhod space D(]0, T], H) and
by E,p the corresponding expectation. We also remark that the o-algebra ff =
0{ Z099 (s ), s < t} C .7}, where {ft }t>0 denotes the natural filtration of the com-
pensated Poisson random measure N(dvds) and o{ Z02‘p(s), s g t} is the o-algebra
generated by { ZOWs) )39.

Let us consider now the solution {Z (r) lre[t,T] of

Z(r)=Z(t)+/t S(T-—-u)F Z()u )IIII+//ES( I—II) B(,IIZIII ))N(dvdu).

From Theorem 3.2.1, it follows that {Z (I) lre[t.T] is stochastic equivalent to
{Z“Z<t)(r) Lear]. Let H(z, t, r, w) :2 Z‘Iz(r), r E [t,T]. We remark that H(z,t,r,co)
is independent of 7-]. Let 'y be a bounded, real valued measurable function on H.

Then we can write
E.III2II + h)) I EII— — EII (11(Z(I w + II III I EI (3.5)

22

and

Ez(t)[7(z(h))l = ElA/(H(Z101h2w))lz=2(t) (36)

where E[']z=Z(t) :2 E[2 | Z(t) = 2;]. We shall prove that
EIIIHIZIII. II + hlel IEI = EIIIHII. I, I + IIIIIIIIEZIII (3.7)
It then follows from (3.6) and the homogeneous property that

E[’7’(H(Z(t),t,t+ h,w)) I ft] = E[’7(H(Z,t,t + hiw))]z=Z(t)
= Elfil(H(ZIOIhvw))lZ'—‘Z(t)

= EZ(t)l7(Z(h))l2

Then use (3.5), it follows Ez[7(Z(t+h)) | 55] = Ez(t)[7(Z(h))] and, since .7212“) Q 7-},
this gives E III2 II + I» I 2.2 ”J— — EII I (Z(h))]-

Proof of (3.7): Put g(z,w) = 7(H(z,t.t + h,w)). Clearly g(z,) is measurable
in w, and z -—> g(z,w) is continuous by the continuity with respect to the initial
condition. Thus g(z,w) is separately measurable, since H is separable. By a theorem
of Mackey [26], we can find a function equal dz 8 dP a.e. to g(z,w), which is jointly
measurable. We again call it g(z,w). Clearly g is bounded. We can approximate g

pointwise boundedly by functions of the form 2:; ¢k(z)ilIk(w).

Ell/(Z (W) ”liftlzn]T;OZ¢k(Z( Efibk (Wllft)
= .3st EI IIIII IIIIII I I 2.1.22 I) — -E[9(2Iw)]I=zII).
k: l

where in the first inequality we used that gbk(Z (t )) is 71- measurable. Cl

23

Definition 3.3.1. For a Markov process {(t), defined on (QT, {ft hag, P) with
state space X let P(17,I,,I‘) = P(§ (I) E F | £(0) = 77) (transition probability function).
We say that P(17, t, F) has the Feller property if for any bounded continuous Borel—
measurable function 05 on x, (P,q§)(77) = fx #7) P(n, t,d7) is continuous in (t, 17) for

t > 0, n E x.
Theorem 3.3.2. The mild solution of (3.1) has Feller property.

Proof. Let h E Cb(H) (bounded continuous functions on H) and let 9% —> Ip, (pm 90 E
H. Then we know Zion(t) —> Z90(t) in probability as n —> 00. Then E[h(Z‘pn (t))] ——>
E [h(Z 90(t))] since otherwise there would exist 5 > 0 and a subsequence, denoted by
n, such that ||E[h.(Z‘Pn(t))] — E[h(Zic(l))]|IH > e and Zipn(t) —> ZWt), a.s., which

yields a contradiction. D

24

Chapter 4

Approximating system

We shall show in this chapter that every mild solution can be approximated by a

strong solution.

4.1 Sufficient conditions for a mild solution to be

a strong solution

We start with a Fubini type theory.

Proposition 4.1.1. Let T = [0, T] and let B : 'II‘ x 'll‘x ExQ ——> H be measurable, and
B(s,t,v) is ft-measurable for each s, and ijijfE E|]B(s,t,v)||§, fi(dv) (1t (13 < 00.

Then

T T ~ T T ~
/ / [B(SItIv)N(dvdt)d.s-:/ // B(s,t,v)dsN(dvdt).
0 0 E 0 E 0

Proof. (Sketch) We first prove that for the B given above, there exists simple functions

25

8,, having the form

—ln-l

WWW = mI;1AII-IWFIIWIIII,III...I“>1III,II+II<3>aw- (41)

'6'

u.
ll

Here A,“ E B(E/{0}) (0 ¢ T),t tjk E (O, T], tJ-I, < tjk+1 sjE (0,T],sj < sj+1,FJ-kz E
ftjkvajkl E H. For allj E 1,...,p— 1 and k E 1,...,n — 1 fixed, Ajkll x ijllfl
Ajklg x ij12 = gb, if 11 74 l2. Also Bn L2-approximating to B w.r.t. ds ® v ® P. The
proof follows almost exactly from Theorem 4.2 in Rudiger and Ziglio [38]. Now for
simple functions in (4.1), we have

fT/O/B ()1vs,I,II (dvdt)ds

p-ln— 1 m

:ZZZ(SJ+1_ SJ )a’JklleH(w w)N((t jka tjk+ll n (0 T] X Ajkzn E)

j===llclll

=/()TA/()TB(s,t,v)dsH(dvdt).

Also by the inequality

B(s, t,i) )ds N(dvdt):

=E/0T/EE E/OB( s(,t,v)ds2
<T/OT /0T fEEHB( (s, t, v )IIHds3(du)dt

we know as n —> oo, w.p. 1, we have

/T/0T/EB( (II')8tL N(dvdt).ds= [OT/E/OTB (3,)tvdsN(dvdt.). [:1

Proposition 4.1.2. Suppose that:

         

 

 

”B(dv) dt

 

 

 

 

(a) Ip E D(A), S(t-r)F(y) E D(A), S(t—r)B(u,y) E D(A), for eachy E H,v E E

and t > r,

26

(C) L; HASH - 7‘)B(I«’Iy)HiIflaw) S 92(t - I“)(1+||y||i1)I 92 E L1(0IT)-
Then a mild solution Z (t) is also a strong solution.

P roof. By the above conditions, we have
T t
f / llAS(t — r)F(Z(r))||Hdrdt < 00, 111.111,
0 o
and
T t
f // EHASU " T)B(‘U,Z(r))||§, [3(du) drdl < oo.
0 o E

Thus by Fubini’s theorem, we have

/0t/03 AS(s — r)F(Z(r))drds
=/Ot/rtAS(s—r)F(Z(r))dsdr
=/OtS(t — r)F(Z(r))dr — [OtF(Z(r))dr.

By Proposition 4.1.1, we also have

jot/08]]; AS(s — r)B(v,Z(r))./V(dvdr)ds
= [Gt/E/tAS(s—r)B(v,Z(r))dsN(dvdr)
= [Oi/E S(t — r>BII Z(r)) NIdvdr) — f/E BII, zIIII NIIIIII.

Hence AZ(t) is integrable w.p. 1 and

t

+ / S(t—r)F(Z(r))dr- / EIEIIIIII

)

f0 AZ(s) ds =S(t)<,a — a

0
III/Esra—I~)B(II,Z(I~))N(IIIIIII)—At/E B(v,Z(r))1V(dvdr)

27

~

=Z(t) — II— Ingmar — fot/E B(v,Z(r))N(dvdr).

Thus

~

Z(t)=Ip+/OtAZ(s)ds+/0tF(Z(s))ds+AtLB(iI,Z(s))N(dLIds).

So Z (t) is a strong solution. C]

4.2 Approximation part
We now study the approximating system, which has the form

dZ(t) = AZ(t) dt + R(n)F(Z(t))dt + /E R(n)B(v, Z(t)) N(dvdt), Z(O) = H(n)ga

(4.2)
where n E p(A), the resolvent set ofA (p(A) := {A E (C: A—A: D(A) —> H is bijective}),
B(n) = nR(n,A), and R(n,A) = fooo e‘"‘S(t)dt. We begin with a theorem on the

mild solution of the stochastic equation.
Theorem 4.2.1. The mild solution of (3.1) is in C(0,T; Lg’mf, P)).

Proof. Let Z (t) be the solution of (3.1). We know that

Z(s + I) =S(s + I) I + / S(s + t — u)F(Z(u)) du

+/0 [38(3 + t — u)B(U, Z(U)) N(dvdU),

Z(s) = 5(8) Ip + foe S(s — u)F(Z(u)) du + [OS/E S(s — u)B(v, Z(u)) N(dvdu).

28

So

Z(s + t) — Z(s) =S(s + t) a — S(s) a + [H S(s + t — u)F(Z(u)) du
+ /s(S(-s + I - I) — S(s — u))F(Z(u)) In
+ f“, [E S(s +t — u)B(v,Z(u))1V(dvdu)
+ / [IEII + I. — I) - S(s — 11))B(vIZ(U))/V(dvd'M)-
We have
E||Z(s +1.) — 2(3)“; 3 5(E11+ E12 + E13 + E]. + E15),

where

Ell =E||S(s + no — swan}, _. 0, as I _. 0,
2

E12 =E [8H S(s + t - u)F(Z(u)) du

 

 

 

 

H

gtE/s ”S(s + I — IIEIZIIIIIIIIII

s+t
gIIIFE/ ||F(Z(u))||§, du —> 0, as t —> 0,

2
E13 =E

 

 

 

 

[03(S(s + t — u) - S(s — u))F(Z(u))du H

SsE/OS “(S(s + t — u) — S(s — u))F(Z(u))||:,Z{ du -—> O, as t ——» 0,

2
E14 =E
H

[H/E S(s + t - 103(2). 2(a)) N(dv du)

 

 

 

 

s+t
31112E/ / ”B(II,Z(a))||§,,a(dv)du —> 0, as t —+ 0,
s E

2

E15 =E

 

 

 

 

f3] (S(s + t — u) - S(s — ullBU)» Z(“ll N(dvdU)
o E

H

SAI2E/ / “(S(t) — I)B(1),Z(’Il))|l%{[3((1’0)(l‘ll.—> O, as t —> 0.
0 E

29

By the fact that ||S(t)|| _<_ M, for Vt E [0, T], ||S(s + t)r — S(s):I:HH —+ 0 as t —> 0,

for VT. 6 I], and Lebesgue dominated convergence theorem. So we have
EI|Z(3 + t) - Z(Slllii -* 0: as t —* 0.

Also by Theorem 3.2.1, we know the mild solution of (3.1) is in C(O, T; L5,“), 15', P)).

E]
The following is the main result of this chapter, which generalizes Ichikawa [17].

Theorem 4.2.2. The stochastic differential equation (4.2) has a unique strong solu-
tion Z(t, n) which lies in C(0,T; L§I(Q,f', P)) for all T and Z(t, n) converges to the
mild solution of the stochastic equation (3.1) in C(O, T; Lf(Q,J-', P)) as n —+ 00 for

all T.

Proof. We know AR(n) is a bounded operator and suppose that |AR(n)l 5 MI.
The first part is an immediate consequence of the existence of a mild solution and

Proposition 4.1.2. Observe, by the growth condition and lS(t)| S e‘“,
IIAS(t - 7‘)R(n)F(y)||H S 60(t—T)A11\/i(1+llyllfl)a
and similarly,
[E IIASII — I')R(E)B(v.y)lliIW1”) s 80““1MEII1 + III/III).
To prove the second part, we consider
Z(t) — Z(t. n) =SII>II — EIIIII + f SII — r)[F(Z(r)) — RIIIEIZIII n))l Ir
+ ff S(t — r)[B(v. ZIIII — EIIIEIv, Z(r. n))I NIIvd-r)
0

30

=/ S(t — r)R(n)[F(Z( r)) — F(Z(r, n))] dr
+/O/ES( (t- r) R([n) B(v, Z(r ))— B(u, Z(r, n))] N(dvdr)
+{S (t)[Ip— R( (n)cp]+/0 S(t—r)[I—R ()(]F ))dr+

+/0/ES t(—)—r[I B(n )(]Bv,Z r))N(dvdr)}.(T

We have,

E sup IIZIs >— ZIs IIIII._ < 3E sup [11+ II + III

OSsSt 0<s<t
where
2

E sup 11=E sup

OSSSt OSsSt

 

 

A. 5(3 — r)R(n)[F(Z(r)) — F(Z(T‘, n))] dr

 

 

.H

t
§4ke2attfo E sup “Z(r )— Z(r,n)||§,ds,( by Lemma 3.2.3)

0<r<s
2
E sup [2 =E sup

0£sSt 053<t

 

 

//3( s—I) II)(,([BIIZ (r))-—B(v,Z(r,n))]N(dvdr)

 

 

11

_<_4kb1/F I sup “Z(r )— Z(r,n)||3,ds, (by Lemma 3.2.3)
o

OSrSs

and

E sup I3 =E sup

ogsgt 0335:

+ [0 /ES(s — r)[l — R(fl)]B('UIZ(T))N(dvdT)

 

 

S(s)[¢ — R(n).,9] + /03 S(s — r)[I — R(n)]F(Z(r)) dr

2

 

 

.H

53E SUP([31+ 132 + 133)-

OSsSt

We now estimate each term in 13, because [B(n) -— [l —> 0, as n —+ 00,

E sup II. = E sup IIEIIIII — EInIIIIIII s |R(n) — IIQE"‘H'~EHiI —» o as n ——» so.

Ogsgt Ogsgt

31

E Sllp I32

OSsSt
2

(by Lemma 3.2.2)
H

 

 

=E sup

OSsgt

[08 S(s — r)[I — R(n)]F(Z(r)) dr

 

 

t
SIEIII — Ilzez"‘tl(t + / EIIEIIIIIII a.) a o as n —» so,

E sup I33

Ogsgt

 

 

f/s ()—I—I~[1 R( (II,)]B(II 2a)) N(dvdi)

 

=E sup

OSsSt

 

H(by Lemma 3.2.2)

§|R(n)—I|2b1l(t+/ E||Z(r)||Hdr) —->Oasn—+oo.
0
Thus we get

E SUD ||Z(8) -Z(8In)||i1 S 64/0 E sup ||Z( )—Z(7‘In)|l31dr+|R(n)-1|265-

OSsSt 0<r<s

Here c4 and c5 are positive constants.

By Gronwall’s inequality, we have
E sup ||Z(s) — Z(sIn)lliI s IEIn) — IIQIIICI‘.

OSsSt

So

0_<_ lim sup E||Z(s)—Z(s,n)||i,£ lim E sup ||Z(s)—Z(s,n)||§,=0. C]

"#00 ogng "“°° OSsST

32

Chapter 5

Stability properties of the mild

solution

5. 1 Ito formula

Let C§(H) denote the space of all real-valued functions It on H with properties:
(i) It'(:1:) is twice (Fréchet) differentiable,
(ii) I/2’(a:): H —+ H is uniformly continuous on bounded set,

(iii) ub"(a:): H —+ L(H) is uniformly continuous in strong operator topology on

bounded set.
By C: ’10C( H ) denote the space of all functions in C§( H) with properties:
(i) There exists m1, such that llw'(:r)||H g mlllrllg, for VT 6 H,

(ii) 1/)” are independent of :13.

33

we recall Ito formula from Rudiger and Ziglio [38].

Theorem 5.1.1. If If)(t,;z:) is once continuously diflerentiable in t, 1/)(t,-) E C§(H)
for any given t, also the second Fréchet derivatives 1,9“, 1/)”, If)” exist and are uni-
formly bounded on on [0, t] X B(O, R) {centered ball of radius R), for all R Z 0. Let

{Z‘p(t),t 2 0} be the strong solution of {3.1), then

III Z“"(t))= IIo III+ [0 wI(8IZ‘p(8))d8

+/0 (1b,, (3, Z” (5)), AZ‘P(s)+F(Z‘p(s)))ds

+/ [[IIs s)+B(v 2"”(8 III

-II s ZIIs II- II Is ers II BIsersIII] EIIvIIs
+ /0 f5 [112(s, Z‘p(s—) + B(v, Z‘p(s))) — ")(s,Z"0(s—))] N(dv ds).

Here wt and 1/2,, are Fréchet derivatives with respect to t and .1: respectively.
We have another form of Ito formula if 1b is independent of t.

Theorem 5.1.2. Suppose 1/) E C§(H) and {Z¢(t).t 2 0} is a strong solution of

(3.1). Then
II(Z‘p(t)) =IIII + [t £¢(Z“’(8)) Is
+/ / {II (Z"°(s +-)B(v ers III — IIers—IIINIIvIsI
where
EIIII = II'III. A2 + FIIII + [lb-”(Z + B(v. III — IIII — II'III. EII. z)>l IIIIII
is the infinitesimal generator of the Markov process given by the solution of (3. 1).

34

Since Ito formula is only applicable to the strong solution, we will use approx-
imating method to study the stability properties of the mild solution to stochastic

equation (3.1). We recall the approximating systems here for convenience.

dZ(t) = {AZ(t) + RInIF(Z(t)) }dt + L; B(‘nlBlv’ 2“» N(dvdt) (51)

2(0) = R(n)go
where n E p(A), the resolvent set of A, and B(n) = nR(n,A). The infinitesimal

generator [In corresponding to this equation is

c,.IIzI =II'IzI As + B(n)F(z))

+ fElIIz + HIIIBII, zII — IIzI — II'IzI. RInIBII. III III)-

5.2 Exponential stability in the m.s.s.

Following Khasminski and Mandrekar [22], we define stochastic stability first.

Definition 5.2.1. Let ZIP(t) be the mild solution of (3.1), we say that it is exponen-

tially stable in the m.s.s. if there exist positive constants c, 6, such that
EIIZ‘p(t)||§, g ()(i—BtHIpHil, for all g: E H and t > 0. (5.2)
The following gives a sufficient condition for exponential stability in m.s.s..

Theorem 5.2.1. The mild solution Z‘P(t) of (3.1) is exponentially stable in. the m.s.s.
if there exists a function 2p : H ——> R and ib E CE‘IOCUJ) satisfying the following
conditions:

Cllll‘llii S M103 Csllrllii (5-3)

35

£‘ll’($) S —c21l)(17) (5.4)

for Va: 6 H, where c1,c2, c3 are positive constants.

Proof. Apply Ito formula to eCQtIlICr) and Z,’f(t) and take expectation, where Zfif (t)

is the strong solution of (5.1). Then we have

eCQ‘EMZflt» - I(Z.“f(0)) = E / scram + c.IIIZ::IsII Is.
Here

aIIsI =II’IsI, As + RInIFIsII
+ fEiIIs: + RInIBII, sII — IIsI —- II'IsI, RInIBII, sIII IIIII
LIIII =II/(s), Ax + F(s»
+ [EIIII + BII. sII - IIsI — II'IsI. BII. III] III).

By (5-4),

C2tl’(r) + Luv/(I) S -£¢(I) + £n'¢'(1‘)
=<W($)s (ROI) - 1)F($)> (5-5)

Wit + R(n)B(vs :c)) - 1/41?) - (ll/(33): R(n)B(vs 10>]
+/.l , l
-lu”(r + B(vs 1)) - 1L‘(flf)- (I) (I), B(v, Jill]

IIdII.

So we have
t
sC2‘EIIertII — IIzrsIIIII s E / 6623(11(Z:f(8)) + 12(Z;f(s))) 1s (5.6)

0

where

1101) = (ti/(h), (ROI) - 1)F(h)).

36

1 (h) ___ f [ [III + RInIBII, h)) — IIhI — II'IhI. R(n)B(v-. h-)>l ] W)
E —IIIh + BII, hI) — IIhI — II'IhI. BII, h)>]

Now, we will prove that the right hand side of (5.6) —> 0, as n —> 00.
First the integral in (5.5) makes sense, because by Theorem 2.1.1, there exists a
bounded bilinear form R2, lel S M’ (a positive constant), such that the integrand

satisfies

|t/I(I + 3003(1). 93)) - 2W) - (It/(x), B(n)B(v, 16))
- W117 + B(v, 1)) - 1W) - (It/(I), B(vsx))l|
=|R2(R(n)B(v, 1)) - l?2(B’(vs:r))|
S|R2(R(n)B(v,17))| + |R2(B(v.$))|
SM’IIR(n)B(v,I)|liI + M'||B(v, I7)||iI
SM”||B(vsI)HiIs
Here It ” is a positive constant. So the integration makes sense.
We know that limnhoo SUPie[0,T] EHZ"°(t) — Zfi(t)||%, = 0 by Theorem 4.2.2 and
IIZ,“f(t)lliI S 2IIZ¥3(t) - Z‘P(t)llis + 2|th(t)lliII we have supn foT EIIZI°3(8)||3I d8 < 00-
By the Schwartz inequality, we get supn fOT E||Z,f(s)|lH ds < 00. So {||Z,‘f(s)|],n =

1, 2,. . . } is uniformly integrable on Q x [0, T] with respect to the measure P x Leb.

Now we prove the right hand side of (5.6) —> 0, as n —> 00.
t
‘E/ 662311(Z:(S)) ds
. O
t
= l3 / ‘3CQ'S<'+/I’(Zfif(s)), (N(n) — I)F(Z::’(s))> Is
0
t
SE / IsCISII'IzrfIsII, IRInI — I)F(Z:f(s))>l Is
0

37

sE ft€023llI’(Zf(s))llHlR(n) — llllF(Z:"(s))||Hds
S|R(n) — IIE ftec2csmlllZf(-8)HH~/l(1 + IIZ:'.°(s)I|H) Is
SMglR(n) — 1| (M3 is a positive constant)

—+0, as n -——> 00.

By letting

Mir) = MI + R(’n)B(vs 13)) - WI) - (W13), R(n)B(vs-"C))

[(1‘) = lf(.’II + 8(1), :1:)) — i/J(;I:) — (if/(r), B(v,:r)),

one has

lE f ec2$12(Z:f(-s))ds

 

<E/o/E ec2S|In(Z ))—I(Z‘f(s ))If3(dv)ds
=E/ [Es IRIIRIIIBIMIIsIII — 32(B(v,Z.‘f(s)))lfl(dv) Is

Here R2 is a bounded bilinear form by Theorem 2.1.1. By the result in Yosida [45],

there exists bounded linear operator C: H -+ H, such that
R2(R(n)B(v» 23(3)» = <CR(n)B(vs 15(8)), R(n)B(v, 25(3)»,

and

R2(B(v-.Z.f(s))) = (CB(v.25(8)).B(v,Z""(-9))>-

For simplicity, let
=E/O /1;6C28(;I‘ )’3( di

38

We have

lE/(;ec2812(Z,‘f(s))ds
(0300301: 25(8)), 3(71)B(Us 25(5)»
SQ
—(CB(v, Z,‘f(s)), 3(1), 23(5)»
-Q (C(R(n)—[)B('Ust(8))sR(n )3 (’0 Z22(s )))
“(03(2), 25(8)), (B(n) — I)B(v, 25(8)»

 

ICHRtn) - IIHB(v, Zf(8))llH|R(n)lllB(v, 35(8))HH

+|C|

 

lBI-v, szsIIllHlRInI — IlllBIv. ZIIsIIllH
tec25m2 n— Ifs 2
SE / [E , I W) Il2llBII,Z.IIIllH
+ m2lR(n) - IIHB( IIII IIllt IIIIII Is
SIR(n)‘IlE/0[E3m2662'SHB(v,fo(s))||§,fi(dv)ds
S|R(n) — IIE / 3m2e222I1 + llZIIIsIlltI Is

_<_M4|R(n) - 1|,

with M4 a positive constant. Thus we know the right hand side of (5.6) —> 0, as

n —-> 00. By the Lebesgue dominated convergence theorem, we have

e22‘EIIZ2IIII S III

By condition (5.4), we have

So

CIEHZWIHII s C3ll¢lliie—62t-

EIIZ”(I)H%< fllIsll2e ‘22t D

39

The function 1,0(x) E 05““(H) and satisfy the conditions (5.3) and (5.4) in the
above theorem is a Lyapunov function. Now we want to construct a Lyapunov func-
tion if the solution Z “’(t) of (3.1) is exponentially stable in the m.s.s..

First, let us consider the following linear case. Suppose F = O and B = Bo is

linear. Then equation (3.1) has the form
dZ(t) = AZ(t) dt + fE Bo(v)Z(t) N(ds dt)

(5.7)
Z(O) = $0-

We assume [80(1) y 2 B(dv) g d l y l2 and the solution of this equation is Z‘p(t).
E H H o

The infinitesimal generator £0 corresponding to this equation is

fiat/2(2) = <¢"(Z)»AZ) +/E[1l'(z + Bo('v)Z) - 10(2) - (15(3), Bo(’U)Z)l adv)-
Theorem 5.2.2. If the solution Z()”(t) of equation { 5. 7) is exponentially stable in the
m.s.s., then there exists a function 100 E Cf’loc(H) satisfying (5.3) and (5.4) with .C
replaced by £0.
Proof. Let
IOIII = f” EllzsIIIlli II + wllIllt. (58)
where w is a constant to be determined later. Since 20‘” (t) is exponentially stable in the

m.s.s., fooo E “25" (0”; dt is well defined and there exists a symmetric and nonnegative
operator R E L(H) (Prato and Zabczyk [6]), such that I0(',0) := f0°° EHfo(t)||';’_, dt =
(RI,0,I0). Hence

“ll/'00P) = (11390.99) + WHsfilliI- (5-9)
It is obvious that 100 E CE’IOC(H) and wllgoll'f, g 100(Ip) g (|Rl + w)|]g0||§, This proves

100 satisfies (5.3). To prove it also satisfies (5.4) with .C replaced by £0, we note that

40

A is the infinitesimal generator of a Co-semigroup S(t) satisfying ”S(t)” g 6‘“. There
exists a constant A (without loss of generality, we assume it is positive) such that

(z,Az) 3 Alle?) (Ichikawa [17]). Hence we have

 

fiwdi=2e~vI+leIoainasIa+smam. use
E
Also,

d r . E¢ Zia 7' —

£09600)=d—(E¢(Ag(r)))|,=o=hm ( o( )) We)
T r-+0 1‘
. 1 '
=fln-j/EwumaI=—wn.

r—s T 0

Therefore

£0100(z) =£o<Rz, 2) + wfiollllli:
g _. “z“; + w(2A + dIIIIHis
={—1+w(2/\ + (1)} ”211%,. (511)

Therefore, if w is small enough, (5.4) holds with [I replaced by £0. This proves the

theorem. Cl

For the nonlinear equation (3.1), to assume zero is a solution, we need to assume
F(O) = 0 and B(v,0) = 0. If the solution Z"°(t) is exponentially stable in the m.s.s,

we can still construct a Lyapunov function as (5.8),

uw=jiEwamhI+mwn.
0

But it may not be in (I'b2‘loc(ll). If we assume it. is in (33'1“(11), we claim that it satisfies

(5.3) and (5.4). Now we prove this claim. Since Z"(t) is exponentially stable in the

41

m.s.s, we assume it satisfies (5.2). Hence

°° c
[0 Ell22ItIlltII s allIllt

for all <0 6 H. Therefore, wllpllfi S 10(Ip) S (g + w)||<,0||§,. This proves (5.3). To

prove (5.4), let (25(99) = f0°° E|[Z92(t)||';’, dt. Observe that
E¢(Z“’(r)) = E f EIllzZ"°<"'>IsIlli l ZIIIIIIs.
0
But by the Markov property of the solution of (3.1), this equals

[0 (”E [E (llzm "lIsIlli I $20)] Is,

where ff = 0{ Z ‘P(t),t S r}. The uniqueness of the solution implies
r (P '
EIllI-Z ("lIsIlt l so -—— E(||Z“’(s+1‘)||i1 l f?)-
Hence
EI(Z‘”(1‘))= / Ellz2Is + sIlli Is = / EIIZWSIIIiIds- (5-12)
0 1'
By the continuity of t ——> E||Z9°(t)||§.,, we get

E¢(Z‘p(7')) - 05(99)

7.

 

LIIII =fIEIIZ2IIIIIl.:o = 1133,

. 1 r .
=11n5--/ E||Z*°(s)ll%ds= —llIllt.
7' 0

r—o

Therefore,

£10m =£¢(Iv) + w£|l0l|31
= _ [I]; + 10(2(I,0,AI,0 + EIIII + [E llEII, IIllt III»

S - llrlliI + 211’All'sflll2q + 1“(2(99.F(s~2)) +/ ||B('l«’,¢)llf1 [30110)-
E

42

Since we assume F(O) = 0, B(v, 0) = 0, using the Lipschitz condition, we get £10(Ip) S
—IIIII%I + mm + NE + kIIIIIli’I-
Hence if w is small enough, 1,0(I0) satisfies(5.4). Therefore we have the following

theorem.

Theorem 5.2.3. If the solution Z"°(t) of (3.1) is exponentially stable in the m.s.s.,
F(O) = O, B(v, 0) = O and Ib(’,0) = fooo E||Z¢(t)|]§, dt is in CE’IOCUI), then the function

10(I,0) constructed above satisfies (5.3) and (5.4).

Since we have difficulty showing ¢(I,0) E 03"”(H), we turn to use the first order
approximation to study the exponential stability in the m.s.s. of the solution of the

nonlinear equation (3.1).

Theorem 5.2.4. Suppose the solution 23’ (t) of equation ( 5. 7) is exponentially stable
in the m.s.s., and it satisfies (5.2). Then the solution Zi"(t) of {3.1) is exponentially

stable in the m.s.s. if

2|lz||H||F( llH+ fllEI E. v) llullB(v Z)+Bo()llufi(dv)<-Z-Ilzlli+ (513)

Proof. Let 1/10(z) = (Rz,z) + wllzllg, as defined in (5.9). Since Z(‘f(t) satisfies (5.2),
]]R||< Since 100(3) 6 CE’IOC(H) and satisfies (5.3), if we can show that 100(2)

satisfies

8 QIQ

5.4), then by using Theorem 5.2.1, we are done. Since

£¢0<3l — £0¢’0(3)
= (106(3),F(z)) + A [1,00(z + B(v,z)) _ 100(z)—(1l’6(:) 819(3)” 5(dv)

” [EIIOII + EOIIIII — IOIII — III/16(2),Bo(vlz)>lfi<dv>

43

=2IIE + III, EIsII

+ /(<(R + wIEIv, 3). EII, III — IIE + IIEOIIII. Bo(v)z)) IIIII
s2IllEll + w)|lzllullF(z)llH

+ IllEll + w) [E llEII. 2) — Bo(11)2|lu|ll3(1uz)+ EsIIIzllH IIIII
=IllEll + III2llzllHllEIlelH

+/ HBO/’72) - Bo(v)2|lul|3(vsz) + 80(10):||Hfi(dv)), (5-14)
E

by (5.11) and the assumption (5.13), [1100(2) satisfies (5.4) if we choose 10 small

enough. E]

The following example shows that the usual Lyapunov function is not bounded

below.

Example 5.2.1. Consider the SPDE

a 2’
‘2 () u

dtu(t,x) = (a —2- + yu) dt +/ uvH(dvdt)
8x IvISl

with initial condition u(0,.r) = Ip(x) E [12(—00, +00) 0 [,1(—00, +00), N is the com-

pensated Poisson random measure.

(92 00
Let H = L2 —00,+00 ,Au = 012—” + 711., B 11,1) = uv, u. = _ 11.2le 1/2.
61:2 00

Now we compute E luf°(t)|2 explicitly. Taking the Fourier transformation of the SPDE,

we get

cyan, A) =(—oz2/\2ii(t, A) + 717(1. A)) dt +/ an, My N(dvdl)
lvlSl
=(—02)\2 + 7m“, /\) dt + / u(t, A)v N(dvdt).

- lvlSl

44

So we have

17(t,A)=§5(A)+/0(—a 22W“) (s,)Ads+/0/( ii()t,A1. N(dvds).
[vlSI

Now using Ito formula on [u(t, A)|2 and taking expectation, we have

E|17(t,A)|2

[@(A )I2+2(— a2A2+7)/ E|u(s, A) )|2ds+/O/v2 )B(dv)E]u(s, A)|2ds

[vlSlv

-A 2 —0122 v2" v t '35 2 s
—lIIAIl +[2I A +7)+/| III I]/ El I,AIl I

v[Sl
So solving this equation, we get

{—202A2 + 27 + f v2 0(dv) }t
lvlSl ,

EWUJHZ = |<79(/\)|26
By the Plancheral theorem, with H = L2(-—00, +00).
WP“, ')|2 =|17“°(tv)|2,

and hence we have

Elma)2 =E|17‘*°(t)]2 = E / leIi,A)l2dA

[@(AHQe M51 dA.

EX}

[.0 {—21.12A2+27+ / '112)3((lv)}l

If we assume 2“) + fv IIS1 1) 213(dv) < 0, then we get

{27+ / v23(dv)}t
Elu“’(t)l2 S We '2'51 -

Hence the solution of the SPDE is stable. But

00 00 1
Eu‘pt2dt:/ AAQ dA.
[0 | ()l _xlII Il 20.,s_(2,+ fl.lsn‘2r’2<d's’>>

Thus the usual Lyapunov function f000 E |u*’(t)]2 (ll, is not bounded below.

 

45

5.3 Stability in probability
Definition 5.3.1. Let Z‘PU) be the mild solution of (3.1). We say that the zero
solution of the equation is stable in probability if
lim P(sup [lZ‘p(l)|]H > e) = 0 for each s > 0.
t

”’IPHH —> 0

Theorem 5.3.1. Let Z “’(t) be the solution of equation (3.1). If there exists a function

10(x) E 03”“(H) having the properties:
(i) clllxllfi S 1,0(x) S cgllxllfi, where c1 and c2 are positive constants,
(ii) ianIHH > 5 10(17): A,- > 0,

(iii) £10(x) S 0, for Vx E H,

then the zero solution of equation {3.1) is stable in probability.

Proof. We first obtain the inequality

,forg06H.

 

10(0)
A

E

P(sup||Z“0(t)||H > e) S
t

To prove this, let 0,5 = {x E H : ||x||H < e},T€ = inf{t : ||Z“’(t)||H > 5}. Now

consider the process Z ‘P(t /\ T E). Using Ito formula on 10(x) and Z}: (t /\ TE) and taking

t/\T

expectation, we have E1,0(Z ,‘f(t /\ T 5)) — 1/I(Z‘p(0) =Ef0 5 £,,l/I(Z fls /\ T€))ds.

N ow using a technique similar to that used in Theorem 5.2.1 and
£,,I,./I(Z;f(s A T.)) _<_ —£10(Z,‘f(s A T.)) + £n'I:(Z,‘f(s A 72.)),
we can get E10(Z"(t /\ TE)) S 10(Ip), so
III) 2 EIIZI’II A T.II > A.P(T. < II.

46

This proves the inequality. Now let ,0 ——I O, and we get the assertion. C]

The function constructed in Theorem 5.2.2 for the linear equation (5.7) satisfies

the condition of Theorem 5.3.1. Hence we get the following theorem.

Theorem 5.3.2. The solution Z? (t) of the linear equation ( 5. 7) is stable in proba-

bility if it is exponentially stable in the m.s.s..

For the stability in probability of the zero solution of the nonlinear equation (3.1),

we have the following Theorem.

Theorem 5.3.3. If the solution Z39 (t) of the linear equation ( 5. 7) is exponentially

stable in the m.s.s. and
2||IllH|lF(I)llH+/E “B(v,I)-Bo(v)IllH||B(vsr)+Bo(v)$l|H13(dv) < wllrrllfI (5-15)

for some w small enough in a. sufficient small neighborhood ofx = 0, then the zero

solution of the nonlinear equation (3.1) is stable in probability.

Proof. Since the solution 23° (t) of the linear equation (5.7) is exponentially stable in
the m.s.s, we define 100(1) 2 (Rx,x) + wllxlfil as in (5.9). By (5.14) and assumption
(5.15), we get £10003) S 0. Obviously, 1,00(x) satisfies the other condition of Theorem

5.2.4. Therefore our assertion holds. Cl

5.4 Exponential ultimate boundedness in m.s.s.

In this section, we study exponentially ultimate boundedness properties of the mild

solution of (3.1). We will give a necessary and sufficient condition in terms of a

47

Lyapunov function for the linear case and use the first order approximation to study

the nonlinear case.

Definition 5.4.1. The solution Z‘P(t) of (3.1) is exponentially ultimately bounded in

the m.s.s. if there exist positive constants c, 6’, M such that
EIIZ‘pWIi; S 66‘0‘Hsollfs + M (5-16)
for V90 6 H.

Definition 5.4.2. The solution Z‘P(t) of (3.1) is ultimately bounded in the m.s.s. if

there exist positive constant K such that
17s.-..Ell22IIIllt s K (5.11)
for V99 6 H.

Definition 5.4.3. A stochastic process {.5 (t), t > 0} is said to be bounded in proba-

bility if the random variables |€(t)| are bounded in probability uniformly in t; i.e.,
supP{ |§(t)| > R} ——> O, as R —+ 00.
t>0
Remarks.
(1) If M = O, we get that the zero solution is exponentially stable in the m.s.s..

(2) It is clear that exponentially ultimately boundedness implies ultimately bound-

edness.

(3) Ultimately boundedness implies bounded in probability (by using Chebyshev’s

inequality).

48

Theorem 5.4.1. The mild solution Z‘p(t) of (3.1) is exponentially ultimately bounded
in the m.s.s. if there exists a function 10 : II —> R, also 10 E Cg'loc(H) satisfying the

conditions:

Clll-Fllir — k1 S W33) S Call-I‘ll?! — lC3 (5-18)
£1,0(x) S —c210(x) + kg (5.19)
for Vx E H, where c1 > 0, 62 > 0, c3 > 0, k1, kg, 133 are constants.

Proof. The proof of this theorem is similar to that of Theorem 5.2.1. Applying Ito

formula to ec2t10(x) and Z? (t) and taking expectation, we get
t
ec2tEI(Z.‘f(l)) —II:1I0II = E / e22Iss + En)I(Zt’(s)) Is
0

where

c210(x) + £n10(x) S —£10(x) + k2 + £n10(x).
As before, we get

t
s22‘EIIZrIIII — IIerOII s f I» Is = 530222” — 1);
0

C2
i.e.,
t
ecztEib(Z,‘f(t)) S 10(I0) +/ ec23k2 ds. (5.20)
0
So
w 2 -C2l 2 , k2 -Cgt ,
ECIIIZ (t)||H - k1 S 6 (CallrllH — 1.3) + C—2(1 - 6 ),
i.e.,

E

 

 

1 _ __ k _ .,
E2I1Illi. sC—[Iss 22‘llIllt—kss C2t+k1+-C—:(1—e 22‘I]
l

c _ k _., k k
3—36 C2‘llIllt—3I C2‘+—‘+—2

(1- e—C'Zt').
Cl Cl C] C1C2

49

So there exist positive constants c, 6 and M, such that
E||Z*°(t)ll31 s cs—2‘llIll's’. + M, for VI 6 H.

So Z “’(t) is exponentially ultimate bounded in m.s.s.. C]
From (5.20), we have the following result.

Corollary 5.4.1. Suppose all the conditions in Theorem (5.4.1) hold except condition

(5.18) is changed to
CIIIIIIiI — 1'01 S 100?)
Then the the mild solution Zf(t) of (3.1) is ultimately bounded in the m.s.s., so it is

bounded in probability.

Remark. The above corollary is a generalization of Skorokhod’s work ([40], Theorem
25, p.70).
For the converse problem, we first look at the linear equation (5.7) and have the

following result.

Theorem 5.4.2. If the solution Z(‘f(t) of equation (5.7) is exponentially ultimately
bounded in the m.s.s., then there exists a function 1,00 6 CE’IOC(H) satisfying (5.18)

and (5.19) with .6 replaced by £0.

Proof. Suppose the solution Z: (t) of equation (5.7) is exponentially ultimately bounded

in the m.s.s.. We suppose (5.16) holds. Let

T
IOIII= f EllEtIsIllt Is+wllIll1.. (5.21)

50

where T is a positive constant to be determined later. First let’s show 100 E Cf’loc(H).

7‘

Let ¢0(:r) = f0 E Z5(I)||§,dl. Using (5.16),

 

 

7‘
IsIsI s / (ce‘o’llrllt + MW 3 gllssll'i + W (MI
0

If ||r||§q = 1, then 900(x) S §+MT. Since 255(1) is linear in x, for any positive constant

k, we have Z§‘(t) = ng(t). Hence I,00(kx) = k2990(.’l)). Therefore, for any x E H,

17 C
90 (x) = ”:1: [2 I0 (——) S (— + NIT) [.1‘ I2.
0 lit 0 “13“}, 9 I Iii

C

9 + MT. Then IpO(x) S c’llxllfi for Vx E H. Let

Let c’ =

T
T(.1;, y) =/ E(Zg(s), 23(9)) (13
0
for Vx, y E H. Then T is a bilinear form on H, and by using the Schwartz inequality,

we get

7‘
lTIsyIl = / EIngsI, EgIsII Is

 

 

7'
sf IEllngs-IlliI2/2IEllZiIsIlltI2/2Is

s (/T ElzaIsIllt Is) In (fTEIIZEI’(s)Iliz Is)

1/2

1/2

)1/2

2900(1') 500(3! 5 CIHIHHHI/“H-

Hence there exists a continuous linear operator C E L(H, H) (Yosida [45] ), such that

T(x, y) = (Cx,y), and “C“ = sup |(Cx,y)| S c'. Since Ip0(x) = T(x,x) =
llrl|H=lvllyllH=l

(Cx,x). So 906(x) = 2Cx and ,0f,’(x) = 2C. Hence 900 E CE’IOC(H) and 100(1)?) E

CE’IOC(H). By (5.22) and the fact that 100(0) 2 uIIIIpHfi, (5.18) is satisfied. By the

ZIIIIli, get

 

 

continuity of t ——> E

(I . . E“ Zipr —E'" '1
£0850(’~P) =2‘;(E?0(er(7’))) =11n'1 MK ()( l) ”)(V’)

r=il T-*0 T

 

r—s

. 1 r (p 2 1 T+T (p 2
=1m5 —— E”Z() (3)”11 ‘15 + " EIIZO(S)H11d5
7" o 7" T
= - llsfi‘lliz + E||23°(T)|li1 S “”93“; + 68—0Tllvlli1 + M

S(-1+ ce-22IllIlli + II.
Therefore using (5.10),

£01000?) =£o<po(r) + 2£ollr||f1

S(-1+ 68””)Hrllfi + w(2/\ + d)|ls0|lfs + M- (523)

Therefore, if T > In 2,16" then we can choose oz small enough such that 100(0) satisfies

(5.19) with [I replaced by £0. E]

For the solution of the nonlinear equation (3.1), if 10(0) = fOT E]|Z"°(s)||§, (13 +

3, is in CE‘IOC(H), we can follow the proof of Theorem 5.4.2 and Theorem 5.2.3

 

 

wl vi

and have the following result.

Theorem 5.4.3. If the solution Z‘r”(t) of (3.1) is exponentially ultimate bounded in
the m.s.s., and (0(90) 2 fOTEIIZ‘P(t)||%, dt is in Cf"°C(H) for some T > 0, then there

exists a Lyapunov function for Z‘p(t) which satisfies (5.18) and (5.19) .

Now, we use the first order approximation to study the properties of exponentially
ultimate bounded in the m.s.s. of the solution of the nonlinear equation based on the

same property of the solution of the linear equation.

Theorem 5.4.4. Suppose the solution Z59(t) of (5.7) is exponentially ultimately

bounded in the m.s.s., and it satisfies (5.16) . Then the solution. Zf(t) of the equation

{3.1) is exponentially ultimate bounded in the m.s.s. if

2l|~2||11||F(~2)||H + [E “B(v, Z) - Bo(v)z||H|lB(vs Z) + Bo(U)Ill11fl(dv) < WIIZHiI + M
(5.24)
for any constant MI and

—ds
1_
W< max ——Ce— (5.25)

C .
8>lng 5+AJS

Proof. Let 100(z) be the Lyapunov function as defined in (5.21) with T > In; such
that (5.25) attains its maximum at T. We just need to show that 100(x) satisfies (5.19).
Since 100(3) = (Cz,z) + wllzllfi for some C E L(ll, H) with [[0]] S 5+ MT and w

very small, following (5.14), we have

[ii/’0“) — £0'100(z)
SIICII + w)(2llzl|H|lF(z)llH + / llEIv. 2) -— EoIIIsllHllEIv. 2) + Bo(v):|lu IIIII)

3(f + MT + w) (II/“sni, + Ml).

Using (5.23)

£11’o(~“)S(—1+ 06””)ll2lliz + w(2A + dlllzllii
+ M + (f + MT + w) (wnzn'i, + All)
= (-1 s. + w (5 + 1111)) llzllt + II2A + I + WIllzllt
C
+ M + (0 + MT + w) Ml.

,. c
Since W satisfies (5.25), —1 + ce‘“ + W(b’_ + MT) < 0, and hence we can choose w

small enough such that (5.19) is satisfied. El

53

Corollary 5.4.2. Suppose the solution 23° (t) of equation (5.7) is exponentially ulti-

mate bounded in the m.s.s., if as ||,0||H —+ 00,

|F(<E)||H =0(l|<e||H)

 

[E llEII. II — EOIIIIllHllEII. II + EOIIIIllH EIIII =oIllIlltI.
Then the solution Z‘P(t) of (3.1) is exponentially ultimately bounded in the m.s.s..
Pr00fl Since as IlsEllH —* 00. llF(10)||H = 0(ll‘PllH) and
[E llEII. II - EOIIIIllHlEII. II + EOIIIIllH EIIII = oIllIlltI.
for any fixed W satisfying (5.25), there exists an K > 0, such that
2llIllHllFIIIllH + [E llEIvII -— EoIIIIllHllEIIII + EsIvIIllHEIIII s WllIlli.

for all [[I0I]H 2 K.

But for “90”,, S K, by the Lipschitz condition,

2llrllu||1”’(<E)||11 + [EIIBWI 90) - Bo(v)<e||H||B(vs r) + Bo(v)90|l11 {3(dv)
sllIllt + ||F(s0)lli1 + fEIllEII. IIllI + llEsIsIIllHI2 EIIII
Sllrllfs +l(1+ llafillfs)+ 21(1 + llrllfs)
SK2 + 31(1 + K2).
Therefore
2|l¢|l11|lF(99)||11 + [3 “3(1), 99) — 30(U)90HHHB(Us 99) + BOWWHH 3011’)
SWllI‘ll'i, + K2 + 31(1+ K2).
The assertion follows from Theorem 5.4.4. Cl

54

Corollary 5.4.3. Consider the following system

du(t)=Au(t)dt+F(u(t))dt+ / B(u(t),v)l\l(dvdt)
E

11(0) =99.

Suppose F and B satisfy Lipschitz and growth condition defined as before and A is

an infinitesimal generator of a (0- -semigroup. Then if the solution {11"°(l) ),t > 0} of

du(t) =Au(t)dt

11(0) =90

is exponentially stable (or even exponentially ultimately bounded), as ||Ip||H —+ 00,

||F(90)HH = 0(llrllu) and

[I IIEI IIIIHEI EIIII=IIIIIII1II

Then the solution of the above equation is exponentially ultimately bounded in the

m.s.s..
Proof. This follows exactly from Corollary 5.4.2. [:1
Example 5.4.1. Let us consider the system,
1 ~
d t=—-A.tdt ——-———~dt INdIdt
u() u() +1+|u(t)| +/Ri (1 ),
11(0) =93,

where u(t) is real valued. By the above argument, we know u(t) is exponentially

ultimate bounded in the m.s.s..

Chapter 6

Invariant measures

We will continue to study the properties of the solution in this chapter. The conditions
for the existence and uniqueness of an invariant measure associated to the solution

are given, and finally an example is given to illustrate our theory.

6.1 Introduction

Let H be a real, separable Hilbert space defined before and B(H) be the Borel o-
algebra. Let Z (t) be a Markov process with transition probability P(t,y, B),t Z
0,y E H, B E B(H). We define T(t): Mb(H) —+ MI,(H), its semigroup, by

[T(t)hl(y) = [H hIzIEIII. Ith e MsIHI,
where Mb(H) is the space of bounded measurable functions on H.
Definition 6.1.1. Let Cb(H) (Cu,(H)) be the space of bounded continuous (weakly

continuous) functions on H. The semigroup T(t) (or the Markov process Z (t)) is said

to be Feller (Iv-Feller) if T(t)(.7b(ll) C (,‘b(ll) (T(t)(§'w(H) C (I'u,(ll)) for I. Z 0.

56

Definition 6.1.2. A sequence of probability measures [in on B(H) is said to be weakly

(w-weakly) convergent to a probability measure )1. if for any h E Cb( H ) (Cw(11))

gig/H thII.IIII= /Hh(yM(dt/)-

Definition 6.1.3. The set M of probability measures on B(H) is weakly (iv-weakly)
compact, if from any sequence of probability measures in M, a weakly (w-weakly)

convergent subsequence can be extracted.

Definition 6.1.4. Let p be a o- -finite measure and let th (2A) fHP (t. y, A) u (dy).
Then [1 is said to be an invariant measure associated to the Markov process if [tTt = u

for all t Z 0.

6.2 Existence and uniqueness of an invariant mea-

sure

We first recall some known results.

Theorem 6.2.1. The set M of probability measures on B(H) is weakly compact iff for

each s > 0, there exists a compact set K C H such that sup{ u(H\K);/1 E M} < 5.

Remark. This is Y. V. Prokhorov’s theorem (Billingsley [2] ).

Theorem 6.2.2. The set M of probability measures on B(H) is weakly compact iff

two conditions below hold.

i) For any 5 > 0, there exists c > 0 such that n{ y : Hy”); > c} < e for all u E M.

57

ii) The series 2):, [826),]2 is uniformly convergent in p for each c > 0 in some

orthonormal basis {eIc }, where [Life]? = L'yIIHSC(z,y)2;1.(Il1/) for z E H.
Remark. This is a result from Gikhman and Skorokhod’s book ( [11] ).

Theorem 6.2.3. The set M of probability measures on B(H) is w—weakly compact if
for each a > 0, there exists a weakly compact set K C H such that sup{ 11(H \K ); u E

M}<e.

Remark. This is Y. V. Prokhorov’s theorem under the weak topology.
The following lemma is also known (Ichikawa [15] ). It will be used often in what

follows. We give the proof here.

Lemma 6.2.1. Let p > 1 and g be a nonnegative locally p-integrable function on

[0, +00). Then for each 5 > 0 and real d

t p t
(/ ed“"’")g(r) dr) S C(Esp) / ep‘d+€’(“")g”(r) dr,
0 o
. 1 1
for t large enough, where C(5,p) = (1 + gs)?” with — + - = 1.

P ‘7

Proof. First, we use Holder’s inequality to get

t t
/ ed(t—r)g(7.) d7. =/ [e(d+e)(t—r)g(r)][e—e(t-r)] (17‘
0 0
1 1
t — t _
S [/ [e(d+€)(t—r)g(r)]P ClT] P [/ [e—e(t—r)]q {17‘} q .
0 0

I)

t P t t
[/ ed(t—r)g(r) (17] S / ep(d+6)(t—r)gp(7,) CIT [/ [e~5(t~—r)]q dr] q .
0 0 0

58

So we have

Observe that
t t 1
/ [e—e(t—r)]q (h. _____ / e—€q(t-r) (17‘ = _(1 _ (3'5“).
0 0 5‘1

So for any given 5 > 0, q > 1, we need to prove

I
EU — 6W) S 1+ (18.

or we need to prove that

e”£qt + qs + q252 — 1 Z 0.

Because when 5 = 0, e‘f"t + q5 + q2e2 — 1 = 0, and

1
The?“ + qe + 11262 - 1) = -qte‘“" + q + 2426 = I1(—tE‘eq + 1 + 2(15):

I
when t is large enough, we have q(——te”€qt + l + 2qe) > 0, and so —(1 — e‘eq‘) S
511

1 + qe. El
The following lemma is from Liu and Mandrekar [25].

Lemma 6.2.2. Suppose Z ”(t) is ultimately bounded. Then for any invariant measure

m of the Markov process Z"°(t), we have

f llylltmIIII s K’ < 00.
H

Proof. Put f(x) = “T“h and f,,(x) = [[0,n](f(x))f(x), where I is a characteristic
function. We note that f,,(x) E L1(H, m). From the assumption of ultimate bound-
edness, there is a constant K such that finfitnoo E]|Z“°(t)||§, S K for any Ip E H. By
the ergodic theorem for Markov process with invariant measure (Yosida [45] ), the
limit

T
lim %/ Ptf,,(x)dt = f,':(x) (1n — a.e.)

59

exists and Emf; = Emfn, where

12.1.0): [H AIyIEIII. dy)s

and Emfn = f” f,,(x) m(dx). From the inequality fn(x) S f(x) and the assumption

of ultimate boundedness, we have

T T
1.:III= urn-71.] manning / EIIIIII

T—soo

T
= E 1/ E|IZ‘I2(i)||§,di s K’.
cor 0
Also from [u(x) T f (x) as n -—> 00 implies that
Emf = lim Emfn = lim Emf; S K'. I]

Now, we examine the uniqueness of invariant measures. Let BR = {y : ||y||H <

R}. The following theorem is from Ichikawa [15].

Theorem 6.2.4. Suppose Z‘l’(t) is exponentially ultimately bounded and for each

R > 0,6 > 0 and e > 0, there exists T0 = T0(R,6,5) > 0 such that
P{ ||Z¢0(t) — Z¢1(t)|IH > I5} < e for any $0,101 E BR whenever t2 T0. (6.1)
Then there exists at most one invariant measure.

Proof. Let 111,-, i = 0,1, be invariant measures. By Lemma 6.2.2, for each s > 0,
there exists an R > 0 such that rn,(H\BR) < e. Let h E Cw(H). Then there exists
T = T(s, R, h) such that |[T(l)h](<,00)—[T(t)h](,:l)| S e, for 1,90. Ipl E BR, iii 2 T. Now
we prove this statement. Let K be a weakly compact set in H. Recall that the weak

topology on K is equivalent to the topology defined by the metric

00

1
III. 2) = 2: ~27 III, II - am. I e K
k=1

60

for any fundamental subset { ek } of H; namely, the closure of the linear span of { ek }
is II. First we shall show that for each n > 0 and e > 0, there exists a T2, such that
t 2 T2 implies
P01429900» - IIZ‘I’IIIIII s 11}21- s

for all 900,901 E 83. By exponential ultimate boundedness, we know there exists T1
such that t 2 T1 implies P[Z990(t) E BR] 2 1 — 8/3 for any ,00 E BR. Note that h on
BR is uniformly continuous with respect to the metric p. Hence there exists a I5 > 0,
such that y, z E BR and p(y,z) S 6 implies ]h(y) — h(z)| S y. Note also that there

exists an integer J such that

Z films—2H g 5, for all y,z 5 BR-
k=J+1

Now choose T2 2 T1 such that t 2 T2 implies
J
E PIIIII. 29200) — 2921(1))13 I/2l 21— s/3
k=1
for all ,00, (01 E BR. This is possible by (6.1). Then for t 2 T2, we have

P{Ih(Z"5°(t)) - h(Zf)1(t))| S 11}

2142200). 221(1) 6 Es,1sIz20III.z21IIII s I}

J
2E{z2OIII. Z21III 6 Es: flex Z20III — z21III>I s 6/2}

k=1

Now for any given 5, choose T such that t 2 T implies

P{ lhIz20IIII — 11(Zf1(t))l 3 s0} 21- 4—K0

61

 

where K0 = sup [h(y)] < 00. Then

EIIIIzs’on»4.1221(0)]<— :+2K0(41:0=) a.

Note that
/ h(y) m,(dy) = / [T(t)h](y)m,-(dy) for i = 0, 1.
11 11
Consider
/ h(y0)m0(dy0)— Lh(yi)m1(dy1)]
= /11/H[h(1— yo) h(yl )]m0( dy0)m1(dy1)]

 

= /11/H [[T(t)h](y0)—[T(t)hl(yl)]m0(dyolm1(dy1)]
s f / [lT(t)hl(yo) — [T(t)hl(y1)] moIIIsI mIIIIII

_ —(+/BR /H\BR) (/312+ /H\BR)I[T (ti—hl(yo) [T()ll](y1)]mo(dy0)1111(dy1)

£8 + 2(2K0)€ + 2K052, 1ft 2 T,

where Ko- — sup|h(y )I < 00. Since 5 is arbitrary, we conclude f” h(y)m0(dy) =

L, h(y) m1(dy), which implies mo = m1. C]

The following Proposition (6.2.1) gives a sufficient condition for (6.1) holds.
Remark. The condition (Ay, y)S ally“2 for y E D(A) is equivalent to [S(t)] S e"‘,a

is real (Ichikawa [17] ).

Proposition 6. 2. 1. Suppose that (y, Ay)S —c0|Iy]]§,, y E ’D(A), and CO is the maxi-
mum value satisfying the above inequality. Also suppose k is the minirnurn value sat-
isfy Lipschitz condition. Then ifa = c0 — 3k > 0, we have E|]Z970(t) — Z¢1(t)|]i, S

(3‘2“‘llp0 — Ipllli, for t large enough.

62

Proof. Let Z‘pl(t) and 2990(t) be two solutions. Then as in Lemma 3.3.1, we have,
2900(1) -— 2121(1)
t
=EIIIII. - I.I + / S(t — sIIEIzroIsII - EIz2lIsIII Is

0
t -s v9903_ vflls~vs.
+/0/EEII )[B(.Z III EIIEIIIINII II

So
HZQOU) — 29910)”?1
£3||S(t)(soo — I.Ill2. + 3|] [0‘ S(t - s)lF(Z"°0(s)) — “2901001618”:
+ 3]] /Ot/E S(t — s)[B(v,Z1‘50(s)) - B(v, Z¢1(s))] N(dv ds)“;
So

EHZQPW) — Ziplmlliw
‘ 2
gas-ZE‘III. — IIIE. + 3E| / “S(t — sIIEIerIsII — FIzrlIsIIIII. Is]
0
t
+ 3/ / E||B(v, 2900(3)) — B(v, Z901(s))||§, S(dv) ds
0 E
—2 t t — (t . 2
S31: CO “,00 - 991]]?! + 3kE(/ 6 CO slIIZVNs) — Z(p1(s)]]H ds)
0
t
+ 3/ kEIIZ‘pMs) — Z901(s)|]i, ds
0
t
sac—222%.. — All. + 3k(1+ 2sI f I’m—21 + 2)“ — 2>EII220IsI — 22' IsIIII. Is
0
t
+ 3k] EIIZ¢0(s) — Z‘p1(s)||§, ds (by Lemma 6.2.1, 6 is small positive)
0

t
gas-2212M. — I.ll2. + 31. / EIIE20IsI — Z2IIsII12. Is
0

t
+3k/ EllZSS0(s) — 2901(3)][31 ds.
0

63

Letting e -—+ 0 and e2(””260 + ()(t _ 3) < 1, we have
E”2900(1) - Z‘plalllh S 36—2603er - villi + 616/0! E||Z‘P0(s) - 2991(9)”; 618'
So By Gronwall’s inequality, we have
EIIZ‘POU) — ZEIIIIII’I’. s 3s-220‘III. — I.ll2. e222 5 34-220 + 6WI. — I.ll2.. D
We consider H with weak topology.

Theorem 6.2.5. Suppose T(t) is w-Feller and that

1 t
2/ EIIZy°(r)l|i1 dr S M(1 +lly0llf1). M > 0 and for any t Z to > 0. (6.2)
0

Then there exists an invariant measure.
Proof. For integers n 2 to, define mn(B) = gfon P(r,y0, B) dr, B E B(H). Then mn
is a probability measure and f” [lyllfi mn(dy) S M(1+ ”yolfii). Hence for each s > 0,
there exists R > 0, such that mn(BR) > 1 — 5, 83 = {y : ||yHH S R}.

By Theorem 6.2.3, {mn }, n 2 to is w-weakly compact and there exists a sub-
sequence, again denoted by mn, which is w-weakly convergent to some probability

measure m0(-). Let h E Cw(H) be arbitrary. Then
/ [T(t)h](y) m0(dy) =nli_noio/ [T(t)h](y) mn(dy)( since T(t) is w-Feller)
H H

= lim (f) [[T(i + r)h](y0)dr

= 33130 (f) / [T(r)hl(yo)dr

n—KXD

1 n
= lim (—) / [T(r)hl(;ljo) dr ( since T(t)/1 is bounded)
n O
=/ h(y) 111.0(Ily).
H
which implies 1110 is an invariant measure of P(t. y, B). E]

64

Now, we drop the assumption that T(t) is w-Feller. Assume instead that A is self-
adjoint and has eigenvectors { e), }, k = 1, 2,. . . , which form an orthonormal basis for
H and eigenvalues

—Ak l —00 as k —s 00. (6.3)

Then the semigroup S (t) has the representation,

00
y=(Ze—Akt (yek, )ek, y E H.
k=1

We have the following result.

1
Lemma 6.2.3. Assume (6.2) and (6.3) hold. Then mt(-) = 7

t
/ P(r,,00,~)dr is
. 0

weakly compact fort Z to > 0 .

Proof. In view of Theorem 6. 2. 2, it is sufficient to show — TT/ 2E [Z See] )dt is
0 k: 1
uniformly convergent in T 2 to, where Zf0(t) = (Z9000), ek) and {ek} is the ortho—

normal basis given in (6.3). We have
t
ZtOIII =e—2‘k’I... + f I‘M" ‘ ”Is... EIZ20II~III Ir
0
t , ~
+ f / WW - The... B(v,Z100(r))) N(dvdr),
0 E
where kao = (ck, 900). Hence
t . 2
Elzt’OIIII2 gas—2112’s. + 3E] / I‘M” — ”Is... EI2‘20Iv-III Ir]
0

+ 3E] A /Ee—’\k(t — 7‘)(ek. B(v, Z990(r))) N(dvdr) 2 (6.4)

 

Now let K be large enough, so that AK > 0. By using Lemma 6.2.1, for any integer

m > 0 we can show,

K+m
Z _1_/‘T e—QAktfifC‘Q (H. < i/T (3_2AKtllY9 H2 (”< _l—lerOllfi.
szT 0 k0 — T 0 #90 H — 2TAK

For the second term of the right hand side of the inequality (6.4), one has

:1%/0T El Ate—Aka — r) (61-, F(Z‘p0(r))) dr 2
<:2:anfTE(fff“)I<II.12‘(ZI2’O(I~)))IInfill,

which by Lemma 6.2.1, and 0 < 6 < AK is,

dt

 

K-I-rn1
<Z_ T/OTEI (1+26)/e2( (“MN xt—T)|(ek,F(Z900(r)))|2drdt

K+m1

<Z_71,/OTE (1+26)/e2( (“1’5“ )(2—T)|(e,.,F(z‘s20(-r)))I2drdi.

Since Ak T 00 as k —> 00, the second term of (6.4) is

T K+m
3%] (1+26)/0-e2( AK+6l(t_T)EZ](ek,F(Z‘p0(r)))|2drdt

k=K

S(1+2IS)/OT/O‘e_( 2()(AK—5 t)(-T)E”F(z<»00(r))]]§,drdt,

 

which, by Fubini theorem, is

2st " )(2’ 2>EIIFIZ20IIIIII2. IIIs

rT
:(:(A1:+::)T f,<1— s- 20.. W -2>IEIIEI220IIIIIII.II

     

(1+26) T .0 2
SW] EIIEIEIIIIIIIHII.

and by condition (A2) and (6.2),

(1+25) T .0
emf 070+ 122 (r)||i.)d1

T
g———; (1: f5?) (1 + f [0 Enzrop-m, dr)
(1+25)1 , 2
5m“ + MI1+ Il.s.ll..II.

66

 

For the third term of the right hand side of the inequality (6.4), one has,

291.2

K+m12

Ee(_)‘k (e ,B(s,z%(r)))1T1(dvdI) dt

     

 

 

—- e_2)“t_( r) e;c v, 9’ r 26 v r
ng/o f/E .. II EI Z°()))l EIIII]II
Kml Tit —2A(t—r), $00 2, , ,
SszT 0 fofEe K ]((.k,B(’U,Z (r)))| S(dv)dr] dt

1 T

s— T] [f] 6‘2”)“"TlEIIB E.IIIZ‘20IIIII.. 13(d1)dr]dt

and by condition (A2), Fubini theorem, and (6.2),

1 T 2 _ . _
ST/ / le 23"“ T)(1+E]]Z¢0(r)]]§,)drdt
0 0

T T
3%] / Isa—22V“ — 1”)(1 + E||2120(I~)|I3,)didr
0 r

 

T
I
si / —(1 — 12—222” — 2>II1+ EllzroIrIIltI .1.
T. 0 2AK
<i/ —I1+EIIE2OI1III2III
—T_ 2—AK ”
—--(1+1 1 [TEHZ’QOO‘HI2 d1‘)
=K2A T 0 ”
Sm“ + M(1+ll900llf1))-
Thus,
K+m:l
=ZK_ T/OTE1[E[Z(190(’ ”(W2
”9:0ll11 (1+ 26)! 2 l 2
<3 '——1 MI . 3—1 All '1
_ 211.. +3,(,K_,)I + I +III.II..II+ 2AKI + I +III.II..II
——>0 as K -+ 00 uniformly in T 2 to. El

Theorem 6.2.6. Assume (62) and (6.3) hold. Then there exists an invariant mea-

sure for the system.

67

Proof. Taking h E Cb(Y) and using Lemma 6.2.3, we can repeat the proof of Theorem

6.2.5. C]

6.3 An example

The following example shows that —/\k 1 —00 as k —> 00 is not necessary.

Example 6.3.1 (Dam storage problem). We consider the semilinear stochastic dif-

ferential equation

du(t) = A7}(t) dt + d£(t), suppose d§(t) = [H uN(du dt)
mm=I€H-

Here A is an infinitesimal generator of a pseudo-contraction semigroup {S(t) }.

Compared to the general case

dZ(t) = (AZ(t) + F(Z(t))) dt + f5 B(v, Z(t)) N(dv dt)
Hm=w€H,
we have F = 0, B(v,:r) = v. The growth condition and Lipschitz condition are

satisfied, so there exists an unique mild solution 77¢(t), such that

t
n‘p(t) = S(t); + // S(t —- s)uN(du ds).
0 H
For any h E H,
(729010) -,77¢2(t),h) = (S(tlm - v2)7h)=<901- $233101»,
so we have,

(ift’l (t) — 7,9920), 12.) —> 0, for any h E I] as 992 —> 9:1 weakly.

68

 

By Ickikawa [15], we know that 7)"°(t) is w-Feller.
Now if |S'(t)| S 65“", ([1 > O), we have

, ‘ ~ 2
E net, 2 3252’” I212 +2 St—s uN(duds
H H 0 H H

 

t

SII-I’WIIIILH/ / IISII-IIIIIIIIIII
0 H
t

S2e‘2’/1‘||¢|I§1+2/0L8-2v(t_s)llulliit3(duld3
t

s2€2mllscll§1+2f e-21’<‘-3>Is/ IIuIItIIII
0 H

_ , 1 _
goemuwmr+fiu—IQWILHIIIMI

SIIWII+M,

where M is a positive constant. So 77¢(t) is exponential ultimate bounded. By
Theorem 6.2.4 and Theorem 6.2.5, there exists an unique invariant measure.
Consider Ae, = —)\e,, Vi with /\ > 0. This shows that —)\k 1 —00 as k —+ 00 is

not necessary.

6.4 Future research plans

First, we will study the SDE’s in Hilbert spaces driven by more general noises, for
example, the Lévy processes. Second, we will study the applications of our theory to

protein folding problem and finance.

69

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IIIIjMIII‘yIII[If]