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THESIS

Illlllllllllllllllllllllllllllllllllllllll

3 1293 01564

This is to certify that the

dissertation entitled

THE ASYMPTOTIC BEHAVIOR
OF STOCHASTIC EVOLUTION EQUATIONS

presented by

Ruifeng Liu

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Mathematics

 

 

V . Mandrekar

Major professor

 

Date August 1, 1996

MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771

 

 

LIBRARY

Mia: higan State
University

 

 

 

PLACE IN RETURN BOX to remove thlc checkout from your record.

TO AVOID FINES Mum on or Moro data duo.

  

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU In An Afflrmutlvc Action/Equal Opportunity lnctltuflon
m

m1

 

THE ASYMPTOTIC BEHAVIOR
OF STOCHASTIC EVOLUTION EQUATIONS

By

Ruifeng Liu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1996

It .(Iibi Di( i

i o {k lint!!- IEAIIILP.

ABSTRACT

THE ASYMPTOTIC BEHAVIOR
OF STOCHASTIC EVOLUTION EQUATIONS

By

Ruifeng Liu

The purpose of this work is to study the asymptotic behavior of the solutions of
Stochastic Evolution Equations. More precisely, we investigate the stability of and
the invariant measures for the mild and strong solutions of the equations.

A sufficient condition for such asymptotic behavior is the ultimate boundedness
of the solutions. In the first part this concept is studied for the strong solution
under coercivity condition with an eye towards applications to stochastic PDE’s. In
fact, under ultimate boundedness, we get recurrence behavior for the solution in the
second part. Finally, we study asymptotic behavior of the mild solution through
approximation by a sequence of strong solutions.

The main technique used is the construction of a Lyapunov function for linear
equation and use it for non-linear equation through first order approximation. This
makes our results applicable to stochastic lPDE’s. We derive asymptotic behavior

specifically for N avier-Stokes, Parabolic Ito and random heat equations.

To: Ming Wu — my wife
and

our parents

iii

ACKNOWLEDGMENTS

My deepest thanks go to Professor V. Mandrekar, my thesis advisor, for his constant
guidance, encouragement and support in all aspects. His patience and thoughtfulness
have made my stay at Michigan State much easier.

I thank my thesis committee members: Professor Bang-Yen Chen, Professor Mar-
ianne Huebner, Professor William Sledd and Professor David H. Y. Yen for their
patience and support.

I thank Professor Habib Salehi, and Professor James Stapleton for their support
in the last year when I stay at Michigan State.

I also thank the support of the Department of Mathematics, and the Department
of Statistics and Probability in the last five years when I experienced growth in

research, teaching and statistical consulting.

iv

Contents

Introduction

1 Preliminaries and Notations
1.1 Nuclear and Hilbert-Schmidt Operators .................
1.2 Hilbert Space Valued Wiener Processes .................
1.3 Definition of Stochastic Integral .....................
1.4 General Stochastic PDE .........................
1.5 Semilinear Stochastic PDE ........................

2 Ultimate Boundedness and Invariant Measures of the Strong Solu-
tion
2.1 Exponentially Ultimate Boundedness and Lyapunov Function

2.2 Ultimate Boundedness and Invariant Measures .............

3 Weak (Weakly Positive) Recurrence of the Strong Solution
3.1 Ultimate Boundedness and Weak Recurrence ..............
3.2 Weak Recurrence and Lyapunov Functions ...............

3.3 Parabolic Ito Equations and Examples .................

4 Stability and Ultimate Boundedness of the Mild Solution
4.1 Exponential Stability in the Mean Square Sense ............
4.2 Stability in Probability ..........................

mO5I-blk

12

15
16
36

42
42
51
54

56
56
63

4.3 Exponentially Ultimate Boundedness in the Mean Square Sense
5 Appendix

Bibliography

vi

65

71

72

Introduction

The purpose of this work is to study the asymptotic behavior of the solutions of
stochastic evolution equations. More specifically, we study the stability of and the
invariant measure for the mild solution and the invariant measure for the strong
solution. In the case of the strong solution, one needs coercivity condition on the
coefficients if the initial value is ”non-smooth”, thus making the results applicable to
stochastic partial differential equations (SPDEs). On the other hand, in the case of
the mild solution, we can dispense with the coercivity condition.

In the case of the finite-dimensional stochastic differential equation (SDE), Won-
ham, Zakai and Miyahara have considered ultimate boundedness of the solution which
guarantees not only the existence of an invariant measure, but also weak recurrence of
the solution to closed bounded, thus compact sets. We study the above problems by
considering ultimate boundedness of the solution of the SPDE. We use the method of
Lyapunov functions for both the mild and strong solutions. This allows us to derive
the known results in a very simple manner.

In the first part of this dissertation, we study ultimate boundedness and the
existence of the invariant measure for the solution of the SPDE. In the work of Khas-
minskii and Mandrekar [15] and the work of Mandrekar [17], the exponential stability
of the zero solution of the stochastic evolution equation was studied and a Lyapunov
function was constructed under this condition. It is clear that exponential stability
of the zero solution implies that the system has an invariant measure degenerate at

zero. Thus two questions arise:

1. What is a general condition (less restrictive than exponential stability) under

which one can construct a Lyapunov function?

2. Can one consider conditions under which a non-trivial finite invariant measure

exist?

Following the ideas of Wonham [27] and Zakai [29], Miyahara [18] introduced the
concept of exponential ultimate boundedness in mean square sense (m. s. s.) and
constructed a Lyapunov function for the finite dimensional case. We generalize his
work to the linear case for stochastic evolution equations and study the nonlinear
case through first order approximation. We also give sufficient conditions in terms
of a Lyapunov function for a weaker concept, namely, ultimate boundedness in the
m. s. s.. This latter concept implies under appropriate condition on the Gelfand
Triplet, the existence of invariant measures of the solutions and can be used along
with the generalization of another theorem of Miyahara ([19], Th. 2) to obtain the
boundedness of the second moment of the invariant measures.

The invariant measures for the mild solutions of stochastic evolution equations
in the infinite-dimensional case was studied by Ichikawa [12] and was systematically
taken up by Da Prato and Zabczyk [6], where the reader can find additional references.
However, we use techniques of Ethier and Kurtz ([8], Ch.IV, Sec.9) to show the
existence of invariant measures for the strong solutions. As special cases, we derive
recent results on the invariant measures for N avier-Stokes equation [1], Parabolic Ito
equation [3] and an improved version for stochastic heat equation([20], [25]), we also
get the existence of invariant measure in the case of multiplicative noise for the random
motion string introduced by Funaki [9]. In fact, we prove the ultimate boundedness
of the solutions in these cases.

The weak recurrence property to a bounded set was studied by Miyahara [18] for
the solutions of the stochastic differential equations in the finite dimensional case.
For the solutions of stochastic evolution equations in a Hilbert space, Ichikawa [12]

indicated that the same theorem held under the same condition as in Miyahara [18].

In the second part, we study the weak recurrence property to a compact set for the
strong solutions of stochastic evolution equations under the coercivity condition in a
Hilbert space. Under appropriate condition on the Gelfand Triplet, we conclude that
the solution is weakly recurrent to a compact set if it is ultimately bounded in m.
s. s. and weakly positive recurrent to a compact set if it is exponentially ultimately
bounded in m. s. 3.. These results extend the work of Miyahara [18], Wonham [27]
and Zakai [29]. Using the results in chapter 2, we can give conditions in terms of a
Lyapunov function for the weak and weakly positive recurrence to a compact set.
The purpose of the third part is to study the stability and ultimate boundedness
of the mild solutions of stochastic semilinear evolution equations. The pioneering
work in the field was done by Haussmann [10] in the linear case and Ichikawa [12, 11]
for the semilinear case. A good exposition can be seen in book of Prato and Zabczyk
[6]. The methods used by them were a direct attack on the problems. In [2], Chow
suggested the use of Lyapunov functions in the study of the stability for the strong
solution. However, this is not appropriate for the mild solution, furthermore, the
Lyapunov function suggested by him for the linear problem in Haussmann [10] is
not bounded below. In [15], Khasminskii and Mandrekar produced the correct Lya-
punov function for the strong solution under coercivity condition and showed that
non-linear problem could be studied through the first order linear approximation. It
was shown in Mandrekar [17] that the sufficient conditions of Ichikawa for mild solu-
tion could be derived through a strong solution approximation. This also led to the
study of stability in probability. We remove in this dissertation the coercivity condi-
tion. Through the strong solution approximation, we study the stability, exponential
ultimate boundedness and stability in probability for the the mild solution. The main
technique is again to construct an appropriate Lyapunov function. Once this is done,
we can exploit the methods developed in [15] and the first part of this dissertation to
obtain results for the mild solutions. As a consequence, we get simplified proofs of

the results of Haussmann [10], Ichikawa [11, 12], Da Prato, Gatarek and Zabczyk [5].

Chapter 1

Preliminaries and Notations

The purpose of this chapter is to provide background material for the subsequent

chapters.

1.1 Nuclear and Hilbert-Schmidt Operators

Let E, G be Banach spaces and let L(E; G) be the Banach space of all linear bounded
operators from E into G endowed with the usual operator norm I] - I]. We denote by E"
and G“ the continuous dual spaces of E and G respectively. An element T E L(E, G)
is said to be a nuclear operator if there exist two sequences {(1,} C G, {901'} C E“

such that T has a representation

Ta: = Z ajgoj(a:), a: E E.

i=1
with

00
Z llajll - Ilwll < 00
i=1

The space of all nuclear operators from E into G , endowed with the norm

HTIII = inflz Haj“ ' ”903'” = TIE = ZWWWH,
j=l j=1
is a Banach space, and will be denoted by L1(E, G).

4

Let H be a separable Hilbert space and let {6,} be a complete orthonormal system
in H. We denote by (~, ) the inner product in H. If T 6 L1(H, H) then we define

trace of T:
tr(T) = 2(T61, 62')-
i=1

Proposition 1.1.1 UT 6 L1(H, H), then tr(T) is a well-defined number indepen-

dent of the choice of the orthonormal basis {ej}.

Note also that

Corollary 1.1.1 UT 6 L1(H, H) and S E L(H, H), then TS,ST E L1(H, H) and
tr(TS) = 17‘(ST) S ||T||1||5||-

Proposition 1.1.2 A nonnegative operator T E L(H, H) is nuclear if and only if

for an orthonormal basis {Cj} on H

00

2(Tej, 6,) < oo.

i=1
Moreover in this case tr(T) = ||T||1.

Because of this fact, we call a nuclear operator a trace class operator in this case.

Now, we introduce the Hilbert — Schmidt operator.

Let E and F be two separable Hilbert spaces with complete orthonormal bases
{6,} C E, {fj} C F. A linear bounded operator T : E —+ F is said to be Hilbert —
Schmidt if

f: ”Tau? < oo
Since '-1

Z ”T6412 = 222 |(Te.-.f.-)|2 = ; IIT‘fij

i=1 i=1j=1
the definition of Hilbert - Schmidt operator, and the number

||T||2 = (X IITe;||2)1/2

i=1

is independent of the choice of the basis {6;}. Moreover “T“; = |]T*||2.

1.2 Hilbert Space Valued Wiener Processes

In this subsection, we will give the definition of a Wiener process on a separable
Hilbert space K.

Throughout this dissertation, we assume that all the random variables, stochastic
processes, probability measures are defined on a probability space (0,}: P) with a
filtration {fchO- Let K be another real separable Hilbert space.

We start with the definition of a Gaussian probability measure on the Hilbert

space K.

Definition 1.2.1 A probability measure It on a Hilbert space (K, B(K)) is a Gaussian

measure with mean m and covariance Q, iffor arbitrary k 6 K and A E B(R1),
u{:c E K: (19,56) 6 A} = N((m,k), (Qk,k))(A),

where N((m,k),(Qk,k))(A) is a non degenerate Gaussian distribution with mean
(m,k) and variance (Qk,k).

Proposition 1.2.1 Ifu is a Gaussian measure on a Hilbert space (K,B(K)) with

mean m and covariance Q, then
(i) fK(k,:c)/t(d:c) = (m,k),Vk E K,

(ii) fK(k1,:r)(/c2,:c)u(da:) — (m,k1)(m,k2) = (Qk1,k2), Vk1,k2 E K.

Proposition 1.2.2 Let u be a Gaussian probability measure with mean 0 and covari-

ance Q, Then Q is a nonnegative symmetric trace class operator on K.

Now we introduce the Wiener process on K.

Definition 1.2.2 Suppose Q is a nonnegative symmetric trace class operator. A K -
valued stochastic process W(t),t Z 0, is called a Q- Wiener process or a Q-Brownian

motion with respect to {ft}t20) if

(2') WW = 0,
(ii) W has a continuous trajectories,
(iii) W(t) is adapted to f, Vt 2 0,
(iv) W has independent increments,
(v) £(W(t) — W(s)) = N(0, (t — s)Q),v t 2 s 2 0.

Since K is separable, there exists a complete orthonormal system {eg} in K, and

a bounded sequence of nonnegative real numbers A,- such that
Q65 = A,e,~,i=1,2,- .. .
We also have a similar decomposition for W(t).

Proposition 1.2.3 Assume Q is a nonnegative symmetric trace class operator. The

following statements hold.
(i) E(W(t)) = 0,Cov(W(t)) = tQ \7’ t _>_ 0,
{ii} E(W(t),k1)(W(s), k2) = (t /\ s)(Qk1,k2),Vk1,k2 E K,
(iii) For arbitrary t, W has the expansion

W) = Edi-mas.-

where

 

are real valued Brownian motions mutually independent on (Qf, P) and the series

in (1.2.3) is convergent in L2(Q,}', P).
On the other hand, we have the following proposition:

Proposition 1.2.4 For an arbitrary nonnegative symmetric trace class operator on

a separable Hilbert space H, there exists a Q- Wiener process W(t),t 2 0.

1.3 Definition of Stochastic Integral

Suppose K and H are two separable Hilbert spaces. In this subsection, we will

construct the following stochastic integral:
t
/ <I>(s)dW(s),t e [0,T]
o

where W(t) is a K -valued Q-Brownian motion with respect to f} as defined in the
last subsection, and (I) is a process with values that are linear but not necessarily
bounded operators from K to H. Let us fix T S 00, and let I = [0, T]

We define the stochastic integral in several steps.

A process <I>(t),t 6 I in L(K, H) is called simple if it takes only a finite number
of values, i. e., there exists a sequence 0 = to < t1 < < tk = T and a sequence
(Do, (1)1, - - - , (1);.-1 of L(K, H )-valued random variables such that (Pm is Jam-measurable
and

<I>(t)=<I>m, for t6(tm,tm+1],m=0,1,~~-,k—l.

For a simple process (I) we define the stochastic integral by the formula:

, k—l
f0 <I>(s)dW(s) = Z <I>m(Wt,,,..M — Wm.)
m=O

and denote it by (I) - W(t),t E I
Now we introduce a Hilbert space K0 = Q1/2(K) of a subspace of K which endowed
with the inner product
°° 1
(k1, k2)o = 2; you, ee)(k2, e.) = (Q’1/2k1,(Q“/2k1).
i=1 '
Let L3 = L2(Ko, H) be the space of all Hilbert — Schmidt operators from K0 to
H. It is also a separable Hilbert space, equipped with the norm

ll‘l’llig = ZlWa-JDP = ZAaK‘I’est-W

i,j::1 i.j=1

= ”‘16?”le2 = tr(‘I’Q‘1"')

where {g,-} with g, = \//\_,-e,-, {6;} and {f,-} are complete orthonormal bases in KO, K
and H respectively. Clearly, L(K, H) C Lg, but not all operators in Lg can be
regarded as restrictions of operators in L(K, H). The space Lg contains genuinely
unbounded operators on K.

Let <I>(t),t E I be a measurable Lg—valued process, we define the norms

Ham. = {E [0’ ”manger/2 = {E/ottr((<1>(s)Q<I> . (5))d3}1/2

forte T.

Proposition 1.3.1 [fa process (I) is simple and |||<I>|||T < 00, then the process <I> - W

is a continuous, square integrable H —valued martingale on [0,T] and
EI<I>-W|2=III<I>|||?, 095T

Remark 1.3.1 Note that the stochastic integral is an isometric transformation from
the space of all simple processes equipped with the norm HI - |||T into the space of all

H — valued martingales.

To extend the definition of the stochastic integral to more general processes it is
convenient to regard integrands as random variables defined on the product space
900 = [0,00) x Q (resp. 97 = [0,T) x Q), equipped with the product o—field:
B([0,oo)) x f (resp. B([0,T)) x 7"). The product of Lebesgue measure on [0,T)
(resp. [0, T])) and the probability measure P is denoted by Pco (resp. PT ).

For the o—field introduced just above, we consider the sub o—field generated by
the adapted simple processes, this sub o—field is called the predictable o—field, we
denote it by Poo (resp. ’PT). It turns out that the proper class of integrands are

predictable processes with values in L2, more precisely, measurable mappings from

(5200,7300) (resp. (QT,’PT)) into (Lg,B(Lg)).

Proposition 1.3.2 The following statements hold:

10

(i) If a mapping Q from QT into L(K,H) is L(K,H)-predictable, then it is also

Lg-predictable. In particular, simple processes are Lg-predictable.

(ii) If Q is a L‘s-predictable process such that |||Q||IT < 00 then there exists a se-

quence {Qn} of simple processes such that [HQ —— (I’nlllT —+ 0 as n —> 00.

Now we are able to extend the definition of the stochastic integral to all Lg pre-
dictable processes Q such that IIIQIIIT < 00. Note that they form a Hilbert space, we
denoted it by N3V(0, T; L3), and by the above proposition, simple processes are dense
in N3V(0,T; Lg), by proposition (1.3.1), the stochastic integral Q . W is an isomet-
ric transformation from that dense set into the space of all H —valued martingales.

Therefore, the definition of the stochastic integral can be immediately extended to

all elements of N340, T; L2).

1.4 General Stochastic PDE

Let (0,17, P) be a probability space with a filtration {fthzoi K a real separable
Hilbert space and {W(t),t Z 0} a K -valued Hinge-adapted, Q—Brownian motion
defined on (0,.7', P).

Let V C; H be two real separable Hilbert spaces such that V Q H is dense and
V H H is continuous, We identify H with its dual space, and denote by V“ the dual

space of V, therefore, we have

VQHQV“.

v- the norms in V, H and V“ respectively, by <-, ->

 

Denote by H ' ”V, II ' NH and || '|
the duality product between V and V". In addition, we assume that for v E V and
v* E H, <v, v*> = (v, v“). The above triplet V C H E V" is called a Gelfand triplet.

Let M 2(0, T; V) denote the space of all V-valued measurable processes satisfying:
(i) u(t, ) is Ft-measurable; and,

(ii) E [J ||u(t,w)||%,dt is finite.

11

We first study the following equation:

f

u E M2(0, T; V)
< du(t) = A(u(t))dt + B(u(t))dW(t) (1-1)
u(O) = 4,0.

 

L

where (,0 6 H, A : V —> V" is an operator with ||A(u)||v. g a1||u]|V,B(u) E
L(K, H) and ||B(u)||L(K,H) S blllullv for u E V, where L(K,H) is the space of

all bounded linear operators from K to H. Here A, B are in general nonlinear, a1, b1

are constants.

For the existence of solutions of the above equation, we need the following crucial
condition:

coercivity condition: 301 > 0, A and 7, such that for Vv 6 V,
2<v, A(’U)> + t7‘(B(v)C.?B"'(v)) S Allvllir — allvllix + ’7, (1-2)
and monotonicity condition: for Vu, v E V,
2<u - '0, AW) - A(v)> + t7‘((B(U) - B(v))Q(B(u) - 3(0)” .<_ All“ - ”Hit,

Theorem 1.4.1 Under the above coercivity condition and monotonicity condition,

equation (1.1) has a unique solution {u“’(t),t Z 0} satisfying
u‘p E L2(Q,C(O,T; H))flM2(O,T; V).

Furthermore, the solution is Markovian {[23], Ch. 3) and the corresponding semigroup
is Feller.

The above solution is called a strong solution.
The major tool to study stochastic differential equation is Ito’s formula, we quote
it here for the ease of reference [21].

Let Q : H -+ R be a function satisfying:

12

(i) \I' is twice (Frechet) differentiable with Q’ and \P” locally bounded.

(ii) \II, \II’ are continuous on H

(1.3)

(iii) For all trace class operators T, tr(TQ’(o)) is continuous on H -—> R.
(iv) If v E V then \IJ'(v) E V, u —> <\Il'(u), v*> is continuous for each v“ E V“.
(v) [I‘ll’(v)||v S C0(1+||-v||v) for some Co > 0,\7’v E V.

Theorem 1.4.2 {Ito’s formula):Suppose \II : H —) R satisfies the above conditions
and {u"’(t),t Z 0} is a solution of (1.1) with u” E L2(Q,C(O, T; H))flM2(0, T; V).
Then

W(u‘p(t)) = use) + f 1: intends + [(wuas»,B(u:(s))dW(s>). (1.4)

where [I \Il(u) = <‘Il’(u), A(u)> + %tr(\P”(u)B(u)QB‘(u)).

1.5 Semilinear Stochastic PDE

When A is a semilinear operator, equation (1.1) is reduced to the following semilinear

stochastic evolution equation on H:

{ du = (Au + F(u))dt + B(u)dW(t) (15)

11(0) = (,0.
where A is the infinitesimal generator of a (Jo-semigroup S (t),t Z 0 on H satisfying

[IS (t)|| S e“" for some real number w, F and B are in general nonlinear mappings

from H to H and H to L(K, H) satisfying the Lipschitz condition:

||F(y) - 17(2)“ + ||B(y) - B(2)“ S dlly - 2”,
||F(y)|| + ||B(y)|| S d(1+llyll)-

for some constant c and all y, z 6 H.

(1.6)

Besides the concept of a strong solution, for the semilinear case, we have the

concept of mild solutions following [11]:

13

Definition 1.5.1 A stochastic process u(t),t E I, is a mild solution of (1.5) if
(i) u(t) is adapted to 5,
(ii) u(t) is measurable and [GT ||u(t)||2dt < oo w.p. 1 and

(iii) u(t) = S(tho + f3 S(t - 8)F(u(8))ds + If S(t — 8)B(u(8))dW(S)

for allt Z 0 w.p. 1.

In general the strong solution is rather stronger than the mild solution, for the

relationship of these two solutions, we have the following propositions: [11]:

Proposition 1.5.1 If u(t),0 g t :5 00, is a strong solution of equation (1.5), then it

is a mild solution.

On the other hand, under some sufficient conditions, a mild solution can be a

strong solution [11]:
Proposition 1.5.2 Suppose that

(a) u(O) E D(A)w.p.l,S(t — r)F(u) E D(A),S(t — r)B(u)k E D(A) Vu E H,k E
K, and t> r,

(b) “145(11— r)FMH S 91(t - r)IIUHagl E £1(0,T),
(6} ”AS“ - 7‘)B(U)|| S 92(t - 7")||U||,92 E £2(O,T)-
Then a mild solution u(t) is also a strong solution.
For the existence of the mild solution of equation (1.5), we have [11]:

Theorem 1.5.1 Let go be To measurable with Ellgollp < 00 for some integer p Z 2.
Under the hypothesis {1.6), (1.5) has a unique mild solution u¢(t) in C(O, T; Lp(fl, f,u; H)).

Corollary 1.5.1 [ftp is nonrandom, then there exists a unique mild solution of (1.5)
in C(0,T; Lp(fl,f,u; H)) for all p Z 2.

14

Without loss of generality, we assume the initial value (,9 is nonrandom throughout
the dissertation.

Since we reduced the solution of equation (1.5) to H, the Ito’s formula has a
simpler form. Let’s see the Ito’s formula in this case.

Let 02(H) denote the space of all real-valued functions \II on H with properties:

(i) \Il(x) is twice (Frechet) differentiable,

(ii) ‘II’(x) and \I'”(x)x1 for each x1 E H are continuous.

By C52(H) denote the space of all functions in 02(H) with the first two derivatives
bounded. We have the following Ito’s formula [11]:

Theorem 1.5.2 (Ito’s formula):Suppose \II E C2(H) and {u‘p(t),t 2 0} is a strong
solution of (1.5). Then

W(t)) = W) + [2 wows + flaws»,B<u<s>>dW(s)). (1.7)
where .C \I’(x) =< \Il’(x), Ax+F(x) > +-.];tr(\ll”(x)B(x)QB*(x)) is called the infinites-

imal generator of equation {1.5).

Since Ito’s formula is only applicable to the strong solution of (1.5), we introduce

the approximating systems:

{ du = Au + R(n)F(u(t))dt + R(n)3(u)dW(t) (1.8)

u(O) = R(n)cp.
where n E p(A), the resolvent set of A and R(n) = R(n,A) = (n — A)“. The

infinitesimal generator Ln corresponding to this equation is A, \Il(x) =< \Il’(x), Ax +
1[3(71)F(€'=) > +it"(‘1’"($)R(n)B($)Q(R(n)3($))')
Theorem 1.5.3 Under the hypotheses of Theorem 1.1, equation (1.8) has a unique

strong solution uflt) in C(0,T; Lp(fl,f',u; H) for all T and p Z 2. Moreover, uflt)
converges to the mild solution u‘p(t) of (1.5) in C(O, T; Lp(0,f,u; H) as n -+ co, i.e.:

lim sup E(Ilu‘p(t)-u‘.’;(t)||”) =0 (1.9)

"Too t6[0,T]

Chapter 2

Ultimate Boundedness and
Invariant Measures of the Strong

Solution

In this chapter we study necessary and sufficient conditions for exponentially ultimate
boundedness of the strong solution of the stochastic evolution equation in terms of
a Lyapunov function. We will explicitly construct the Lyapunov function in the
linear case and derive sufficient conditions for the non-linear case through the first
order approximation. We also will give conditions for ultimate boundedness of the
solution of SPDE’s and study the problem of the existence of invariant measures and
their second moment. As application of our general result, we obtain recent results

mentioned in the introduction.

15

16

2.1 Exponentially Ultimate Boundedness and Lya-
punov Function

In [15], exponential stability in m. s. s. of the zero solution of (1.1) was considered
and in the linear case a Lyapunov function was constructed. This function was then
used to consider the stability through the first order approximation, in the nonlinear

case. Following [18], we define

Definition 2.1.1 The solution {u‘p(t),t Z 0} of {1.1) is exponentially ultimately

bounded (in [I - Hg) in m. s. 3. if there exist positive constants c,fi, M such that
Ewe)“; s ce'fi‘nsoni, + M. for V80 6 H. (21)

Remark 2.1.1 If [W = 0 we say that the zero solution is exponentially stable in m.

SO SO.

Theorem 2.1.1 Consider equation {1.1) satisfying the coercivity condition (1.2),
and let {u‘p(t),t Z 0} be its solution. If there exists a function A : H —+ R which

satisfies the following conditions:

(i) condition (11.3),

(i) cutout — k. 3 Am 3 canton}. + k3, via e H,
(iii) LAW) S —c2A(<p) + 122, v99 6 V,

where c1(> 0),c2(> 0),c3(> O),k1,k2 and k3 are constants, then {u¢(t),t Z 0} is

exponentially ultimately bounded in m. s. s.

Proof: Since A(cp) satisfies (1.3), apply Ito’s formula (1.4) to it and take expecta-

tion, we get

E t£A(u“’(s))ds

t!

S j;(—c2EA(u‘p(s)) + k2)ds

EA(u‘p(t)) — EA(u‘p(t')

17

Let Q(t) = EA(u“"(t)) and use the fact that Q(t) is continuous in t we have

QIU) S —Cg(p(t) + k2.

Hence
I: k
@(t) s —2 + (M) — ire-:22
C2 C2
i.e.,
k
BMW» 3 —2 + (A(so) — E)
C2 62
Using (ii), we have
tp 2 <p k2 k2 -c2t
clEuu (any — a 3 mm m) s 32— + (Aw) — 3):: (2.2)

|/\

k k
l +(03ll80lli1 + k3 - l)?”-
C2 C2
From the above inequality we get
Ellu“’(t)ll§; s ce‘fi‘llrlliz + M for V‘P e H,

for some constants c, B and M. So u‘p(t) is exponentially ultimately bounded in m.

s. s.
We note that (2.2) gives

Corollary 2.1.1 IfA : H —> R satisfy (i), (iii) in Theorem 2.1 and

W’ clllrlli; — In S A(cp) V99 6 H

for some constants c1(> 0) and k1, then

1 k
limsupEnuru)“; 3 —(k1 + 1).
t—v-t-oo C1 Cg

If {u‘p(t),t Z 0} satisfies the above condition, we say it is ultimately bounded in m.
s. s.. The function A(cp) defined above is called a Lyapunov function. We now will
construct a Lyapunov function if the solution of (1.1) under coercivity condition (1.2)

is exponentially ultimately bounded in m. s. s..

18

Suppose the solution {u‘p( t),t Z 0} of (1.1) is exponentially ultimately bounded

in m. s. s., i.e., we suppose (2.1) holds. Let

A(so) = /OT(/0‘ Ellu‘”(s)lli»ds)dt (2.3)

where T is a positive constant to be determined later.
Applying Ito’s formula (1.4) to ”99”th taking expectation and applying coercivity
condition (1.2), we get

t
/ ELIIu:(sIIItds
0

t t
A] Ellu‘”(s)llisds—a/ Ellu‘°(s)lltds+7t
O 0

Elluw(t)||§; -- llrlliz

|/\

hence
t 1 t
[0 EIIu:<sIIItdss;(A/o EIIu:(sIIIids+IIsoIIi+vtI.

Since {u‘p(t),t 2 0} is exponentially ultimately bounded and it satisfies (2.1)

t 1 c/\ _
f0 Ellu‘p(s)lli/d8 s ——',,—'(1—e fi‘)l|rl|t+llsplliq+IAIMt+7tl
c/\ 7+ /\ M
3 (',,+1' 399)“ IIt+ —i'—t. (2.4)
0/3 0
Therefore,
A(so) = /1(/ Ellubllvdsdt
c__IAI s+IAIM .
< —— . .
_ (0+ 0, )TH IIH+ ,0 T (25)
Now forvEV,
Lllvlfil = 2<v, A(u)> + tr(B(v)QB’(v))
SO

l£llvl|ill S 2aillvlli/ + llB(v)llL(K,H)tr(Q)
2Gilllvlliz + bitT(Q)||v||iz

S C'll'vllix

|/\

for some positive constant c’.

hence

£||vlliq

Therefore, we have

Ellu‘”(t)||§1 - llsolliz

19

Z -C'llv||i/o

()llHdS

/ Brutus

2 —c / Ellu‘”(s)lli/ds
0
hence,
t
6/0 Ellu“’(s)lli/ds 2 IIsoIIt—EIqutMIH
2 llrlIZr-M-ce'fi‘llrlliq
= (l—ce‘f‘leolliI-M
therefore,
T t 2
Me) = [0 (f0 EIqusIIIVdsIdt
1 T
2 —I/ llsolli1(1—ce"")dt-MT]
_ MT
= —[T——(1—e”‘)]llsollH—7
1 MT
> — —— —— .
_ do" gIIIspIIH C. (26)

this proves (ii) if T > 29,-.

Now we need the following lemma to continue:

Lemma 2.1.1 [ff 2 0, and f E L1[0,T] for any T > 0, then

 

T t+At
11m / ft f($)ds
APO 0 At

T
0

ftt+Atf( (8)618
m

11 At

At—vO

 

dt= [T f(t)dt.

Proof: We are going to use Fubini theorem to change the order of integrals:

 

/T tt+m f(3)d3

dt
At

20

= 31.; [OT( /. ”m f(s)ds)dt

= £7[/()At(/sf())sdtds+ S :Atf())sdtd:+/T+At(/:mf(8)dt)dsl
= fivomsfl) (s)ds+/T( f(s) )Ade/M s()T+At—s)ds]

g 517w 0 f(s)ds+AtAtf(S)d3+At/:T(M f(s)ds]

= AAtf(s)ds +A:f(s)ds +/TT+Atf(s ds

the first and the third term go to zero as At —+ 0, so

lim T tt‘l'At f(S
At—rO 0 At

 

Salt 3 /0T f(t)dt.

the other direction of the inequality follows from Fatou’s lemma easily. This proves

the lemma.

Let’s now suppose that A(ip) satisfies (1.3). To prove the converse of Theorem

2.1, it remains to prove (iii). Observe

WE/‘/E( WI"%’IIW(»ea
But by the Markov property of the solution of (1.1), this equals
T t ,p
AARMWWWWEWW

where .7: = o{u“’(r), r S r}. The uniqueness of the solution implies
E(lluuw ”(8 )llvlfl‘)= E(IIUWS + Tlllvlfll

Hence

EA(u“’(r))— [T (/ E||u“’(r+s)||%,ds)dt- /T (/m E||u“’(s)||vds)dt.

Therefore,

£A(so) = %(EA(u:(r)))I.=H

21

so _
z um EM“ (7')) EMSO)
r—+O r
= mg L EW(MH® hEW(flM&fi
r—>O 0 7'
tr+t ‘P 2 _ r ‘p 2
=nm/ EW<8NME.hflwme%t
0 7'

H'tE ‘9 d
= lim/OT ||u (8)”V 3d Edi/0 Ellu‘p(s) Ilvds

r—+0 1‘

= 11 + 12 (2.7)

 

 

r—->0

 

From the above lemma and (2.4)

T r-H ‘p 2
,1: “mt EW(WW®

o r-+0 r

T
=,/Emummu

s<jM+ HmHH+

 

,\ M
+fi-L—l—T (2.8)

and since V <—-+ H is continuous, there exists a positive constant 010, such that Hull}, 5

aollvll'f/ for all '0 E V, so
12 3 —lim——/ E||u (s)||Hds

and by the continuity of the map 3 —» Ellu‘p(s)II§, we have 12 S —aloll99llii

therefore,
ClAl 1 2 7 'l‘ lAlM
A < —— — — —— .
£(W—(ar+a 0mmm+ a 1" on
Let T > ao(%%l + i), then 63L} + i — % < 0, then we get (iii)

Up to now, we have proved the following theorem:

Theorem 2.1.2 Consider the equation (1.1) satisfying (1.2),let the solution {u‘P(t),t Z
0} of it be exponentially ultimately bounded in m. s. 3.. Suppose

w] (fWEW )Iwow

satisfies condition (1.3). Then A(cp) satisfies the conditions in theorem 2.1, i.e., there

exist constants 01(> 0),c2(> 0),c3(> 0), k1, kg and k3, such that

22

clllcplllq — k1 S A(so) S Csll‘Pllii + ’63- for W) E H

and

LAW) S —CgA(cp) + kg. for V99 6 V

Remark 2.1.2 In addition, ifs —> E||u“’(s)||¥, is continuous for W E V, then [2 =
—T||Lp||%,, using {2.8) and by the fact that Hull}, 3 ao||v||¥, for all u E V, we have

CIA 1 7+ /\ M
11<ao(a—,,'+—)u 115+ Ll T
therefore,
ClAl 1 T 2 '7 + lAlM
< __ _ _ _ _—
51W)- ao( a, + a a0)” ”V + a T few 6 v (2.10)

IfT > £1063} + 5;), then £351 + i — 0% < 0 and use the fact “12”}, S aoHle, for all
v E V again, we also have

CIAI + 1
afl (1
Unfortunately, we do not know at this moment if A(go) = fOTUOt E ||u‘p(s)||%,ds)dt

7+ IAIM
a

£A(90)S( T forcpEV

T 2
— a—O)II<PHH +

satisfies condition(1.3) or not. Now we will restrict our consideration to the following

linear SPDE: l
u E M2(0, T; V)

l. du(t) = Aou(t)dt + Bou(t)dW(t) (2.11)
\ u(O) = 90.

We suppose the linear operators A0, Bo satisfy the same conditions as A, B in equation

 

(1.1) with the same constants and the same coercivity condition, that is:
2<v, on> + tr(BonBav) S Allull}, — allvlfi, + ’7. (2.12)
Denote the solution of (2.11) by {ug(t),t _>_ 0} and let

T t 2
Ao(c,o)=/0 (f0 I|u§(s)|lvds)dt for some T.

23

Theorem 2.1.3 If {ug(t),t 2 0} is exponentially ultimately bounded in m. s. 3.,
then A0(<p) defined above satisfies condition {1.3).

Proof: Suppose {u§(t),t Z 0} is exponentially ultimately bounded in m. s. s. and it
satisfies (2.1). From the computation of (2.5), we have

1 c__|)\| 7 + IAlM 2
< _
A002) _. (0+ 7W” IIH+ ——,a T
If llsolliz =1,then w w
c 7 + M 2
< _
A0($0)__(a+ 05W“ 'l" 20 T

Since ug(t) is linear in «p, for any positive constant K, we have

ut’m ) = Isuzu)
Hence,

A0(=k90) szOW )
Therefore, for any up 6 H

+C|/\|
afi

Let c" = (}13' + EL§)T + Zilngz then Ao(cp) S c”||go||%, for V90 6 H. And let

7 + IAIM

2 2
—)T+ ,0 Tlllsclln-

AW )= ll‘PllHA d“: “H )_l(%

T(cp,2/) )2]: (f0 E<u0( 3,) u0(s )>vds)dt for 90,11) 6 H

Then T is a bilinear form on H, and by using Schwartz inequality, we get

ITO/WM = | fl]; E<ut(s),ut(s)>vds)dtl
hT(/ot(E“uB°(S)Il%)%(Ellut(s)Il%)%ds)dt
/T(/tEIIUS( (8) ))lli/ds i(/tE||u$(s)||§,ds)idt

(/T (f. Ellu3(s) )llvds)dt)‘2' (fr ([0 EIIU‘é’(s)IIvds)dt)%
Ao(so ) Aow)

S C"||<p||H - ll¢||H~

|/\ |/\

|/\

24

Hence there exists a continuous linear operator C : H -—> H, such. that

TOWN = (090%), (2-13)

and

llCllL(H,H) = sup [(090, WI S C"
ll¢llH=lvll¢IIH=l

Since A0(90) = New?) = (090,99). so
ABM = 2090 and Able) = 20

Hence, A0, A2, and A3 are locally bounded on H , A0 and A6 are continuous on H and
|A0(90)| S IICIIL(H,H)||<P||in and

||A6(90)IIH = ||20<P||H S leCllL(H»H)ll80llH

This proves (i), (ii) of (1.3). And for trace class Q, QA” = 2QC and trace(QC) being
constant is continuous, hence (iii) of (1.3) holds.

To prove (iv) of (1.3), we observe |T(90,¢)| S Cnll‘PllHll‘pllH and “90”}: S aollellb
for any (,0, 2b 6 V. This implies |T(cp, 2b)| S d’aollcpllvllzbllv. Because T(<p, 1b) is bilinear

on V x V, there exists a continuous operator 6' : V —» V, such that

New) = (590,110 for all at e v. (2.14)

Hence A’(c,o) = 2690 E V for cp E V and cp —+ CVp is continuous on V —> V. Since

IIA’(90)||v = 2l|5<pllv S 2llélllL(v.V)||<r°||v S ZlIéllL(V.V)(ll‘PllV +1)

for any Lp E V, therefore, we have proved A satisfies (1)) of (1.3) and proved the

theorem.

Therefore, we have the following theorem:

Theorem 2.1.4 Consider the linear equation(2.11) satisfying (2.12). Its solution
{ug(t),t Z 0} is exponentially ultimately bounded in m. s. s. if and only if there

exists a function A : H —¥ R satisfying

25

clllrlli; — ’91 S A(SO) S CBll‘Pllii + ’53: W’ E H
and

50/10?) S ‘02A(<P) + ’92, V80 6 V
for some constants c1(> 0),c2(> 0),c3(> 0),k1, kg and k3. where
I 1 II i
50AM?) = <A (9°), A0(9°)> + '2't7‘(A (5P)30(90)Q30(90))-

Furthermore, if we set To = (”(20% + i) + Ea then A0(<p) = 0T°(f(f E||ug(s)||%,ds)dt

will be such a function.

Now, we will consider the nonlinear case by linear approximation in case (2.3) does
not satisfies (1.3). where {u‘p(t),t 2 0} is the solution of the nonlinear equation (1.1).

We need the following lemma ([24], PP. 39):

Lemma 2.1.2 Suppose X is a Hilbert space, T E L(X,X) is a trace class operator.
Define 7'(T) = tr(TT“)i, then it has the following properties:

0) ltrlTll S 7(T),
b) T(TS) _<_ “SHT(T) and 7(ST) S ||S||T(T) for all 5' E L(X,X).

Theorem 2.1.5 Suppose the linear equation (2.11) satisfies coercivity condition (2.12)
and its solution {ug(t),t Z 0} is exponentially ultimately bounded in m. s. 5.. Let

{u"(t),t 2 0} be the solution of the nonlinear equation (1.1). Furthermore, we sup—

pose A(v) - A01) 6 H for all u E V. Iffor v E V,

2Il'vllhrllr‘llv) - onllH + 7(B(v)QB‘(v) - BonBé‘v) S W|lv||i1 + ’5 (2-15)

with w, k constants and

C

w < .
aozsl<1+ 51%)(1+ 5.1% + g) + 551%%(§+ 51% +%)51

 

(2.16)

Then {u‘p(t),t Z 0} is exponentially ultimately bounded in m. s. s.

26

Proof: Let Ao(<p) = 0T°(f0‘ E||u§(s)||%,ds)dt, where To = ao(%l%l + i) + g, then Ao(<p)
satisfies (1.3), and

CllMli] — ’91 S Ao($0) S €3ll<Plli1 + ’93- for V99 E H
for some constants 01(> 0),03(> 0), In, and k3 . It remains to show
£Ao(99) s —62A(Lp) + k2.f0r v 90 e v
for constants c2(> 0), k2. Since A(cp) — Aocp E H, we have

CAoW) — £OA0("P)
= «am, An) — A...» + gtrmzoxBonBiw) — BorQBSrD

= (Assam/1o) - A080) + gtrmzwaoanw) — 8.1008590»

with A6(<p) = 2C<,o,and Ag(c,o) 2 2C for (,0 E V, where C as defined in (2.13) is a

bounded positive operator from H to H, and

7+ WM

T2
20 0

1 clAI
< _ _
||C||L(H,H)_(a+ afl)T°+

with To defined as above. Hence,

Mob?) - £vo(99) = 2(099. A(r) - Aer) + tr(C(B(<p)QB*(<p) — BorQBSrD

Using the above lemma, we obtain:

CAoW) S £vo(99) + 2llCIIL(H.15I)||90||H”A(r) - AorllH
+T(C(B(so)QB‘(so) - BorQBSrD
S £vo(90) + ||C||L(H.H)(2||5P||H”A(SO) - AoWIIH
+T(B(90)QB*(<.0) - 3090623590»

From the computation of (2.9), when T = To,

+ /\ M
comm s —iuapni. + 1—'—'—T.
005

27

therefore
c ’7 + I/\ M
£Ao(‘P) S *3—fillwlli1 + ——|—'To + llCllL(H.H)(wll‘P|llI + k)
0 a
c + A M
s (70-, + wucntwnuwut + kIICIILwn + Lit-LT.

C

Since —aofi + wHCHLmfl) < 0 when w satisfies (2.16), we get the required inequality.

 

This proves the theorem.

Corollary 2.1.2 Suppose the linear equation (2.11) satisfies coercivity condition (2.12)
and its solution {u3(t),t Z 0} is exponentially ultimately bounded in m. s. 3.. Let

{u"’(t),t 2 0} be the solution of the nonlinear equation {1.1). Furthermore, we sup-
pose for v E V, A(v) — Ago 6 H, and

”A(v) - onllh + T(B(U)QB‘(v) - BonBSU) S K(1+||v||i1)
for some constant K > O. Iffor v e V, as ||v||H —> oo
”A(v) — onHH = 0(HUIIH) and T(B(v)QBi(v) — BonBEU) = 0(llvllii)
then {u‘p(t),t 2 0} is exponentially ultimately bounded in m. s. s.

Proof: By theorem 2.5, we just need to show that (2.15) holds for some constants to

and k with to satisfying (2.16). Since for v E V, as Hull” —+ co.
”A(v) — Ao’UIIH = 0(||5’||H) and 7(B(v)QB*(v) - BonBSU) = 0(llvllii)
For any fixed (.2 satisfying (2.16), there exists an R > 0, such that
2llvllherv) - onlln + T(B(U)QB'('U) - BonBé'v) S wllvllii
for V1) 6 V and IIvHH Z R. For v E V but IIUIIH S R, by assumption, we have

2llvHHIIA(v) - AM!!! + 7(B(v)QB'(v) - BonBSU)

S llvllii + ||A(v) - onlliz + 7'(B(v)QB"‘('v) - BonBS'vD
s llvlliz + K(1+||v||iz)
S K + (K +1)R2

28

Therefore, for Vv E V
2II’UHHII/Kv) - onllH + T(B(U)QB'(U) - BonBgv) S wllvlll‘l + (K +1)32 + K
This proves (2.15) with w satisfying (2.16), thus the assertion holds.

Theorem 2.1.6 Suppose the linear equation (2.11) satisfies coercivity condition (2.12)
and its solution {ug(t),t 2 0} is exponentially ultimately bounded in m. s. s., fur-

thermore, we suppose t -—> E||u3(t)||¥/ is continuous for all go 6 V. Let {u‘9(t),t Z 0}

be the solution of the nonlinear equation (1.1). Iffor v E V,

2ll'U||v||A('v) - onllv- + T(B(v)QB'(v) — BonBEIv) S wllvllb + ’5 (2-17)

with w, k constants and

C

to <
(ao + 1W1 + 51%)(1+ 51% + g) + 551%M(§ + 5.1% + gr]

 

(2.18)

then {u‘P(t),t Z 0} is exponentially ultimately bounded in m. s. s.

Proof: The proof is similar to that of the above theorem. Let

Ao(<p) = 0T°(f0t Ellug(s)||%,ds)dt, where To = 006.1% + %) + %. We just need to show
£Ao(<,9) S —czA(<,o) + kg for V (,0 E V

for constants c2(> 0), k2. Since

£Ao(90) — £vo(<.0) = <A6(so), A(r) - A080)> + étr(A3(r)(B(r)QB‘(so) - BorQBSrD

with A6(go) = 25w,and Ag(cp) = 20 for 90 E V, where C" and C as defined in (2.14)
and (2.13) are bounded positive operators from V to V and from H to H respectively,

and

7 + IAIM
2a

7 + IAIM
20

~ 1 c A
”cum. 3 ..,“; + ¢i_d)T° +

1 c|A|
< _ _
||C||L(H.H) ._ (a + afl )T0 +

T02),

T02

29

with To defined as above. Hence,

£Ao(<.0) — £vo(99) = 247% A(r) - Aor)> + tr(C(B(r)QB‘(sO) - 8090623390»-

Using lemma 2.2, we get:

£Ao(9°) S £vo(90) + 2||C~7||L(V.V)Hrllv”A(se) — Aorl
+T(C(B(r)QB‘(r) - 3090623590»
£01100?) + 2H6HL(V.V)”99“VHA(5P) - AOSOHV‘
+IIC||L(H,H)T(B(90)QB*(90) - BorQBSso)

30AM?) + (llélluvy) + ||C||L(H.H))(2||9°||VII/1W)- A0901
+T(B(s0)QB'(99) - BorQBSrD-

 

v.

|/\

|/\

 

V0

Since t —-+ E||ug(t)||f, is continuous for all so 6 V, from the computation of (2.10),

when T = To,
7+ |A|M

c
£01106?) S "Elltolli/ + To,
therefore
c 7 + A M ~
£Ao(90) S —Ellrllb + _b—LTO + (||C||L(V.V) + llCllL(H.H))(wll‘Plli/ + k)

6 ~
S P3“ + W(llClluvy) + llCllL(H,H)))ll90lll/

~ + A M
+klllCllL(v,v) + nanny...) + ”—17%“.

Since —§ +w(||5'||L(v,v) + ||C||L(H,H)) < 0 when (.0 satisfies (2.18), we get the required

inequality. This proves the theorem.

Corollary 2.1.3 Suppose the linear equation (2.11) satisfies coercivity condition {2.12)
and its solution {u§(t),t Z 0} is exponentially ultimately bounded in m. s. 3., fur-
thermore, we suppose t -—+ E||u3(t)||%, is continuous for all (,0 E V. Let {u“’(t),t Z 0}

be the solution of the nonlinear equation (1.1). Iffor v E V, as ||v||v —i 00

“A(v) — onl

 

v~ = 0(IIUIIV) and T(B(v)QB‘(v) - BonBé'v) = 0(llvllzv)

then {u‘p(t),t Z 0} is exponentially ultimately bounded in m. s. s.

30
Proof: By theorem 2.6, we just need to show (2.17) holds for some constants w,k
with to satisfying (2.18). Since for v E V, as ||v||v -—) oo,
”A(v) — onllv' = 0(IIUIIV) and T(B(v)QB’(v) - BonBS’U) = 0(llvlli/l
For any fixed w satisfying (2.18), there exists an R > 0, such that

2||v||v||A(v) - onl

 

v- + T(B(U)QB‘(‘U) — 30052330) S wllvlli/
for all Ilvllv 2 R. By the assumption,

||A(v)|

 

v-. llevllv-Salllvllv and

||B(v)||L(K,H,, llBovllL(K.H) S blllvllva

thus for v E V and Holly S R,

2llv||v||/1(v) - onl v- + T(B(v)QB‘(v) - BonBB‘v)

S 2||v||v(|IA(v)| v- + llevllv-) + T(B(v)QB‘(v)) + T(BonBE§v))
S 4a1||vlllx + llB(v)llT.(K,H)T(Q) + llBOvllL(I\',H)T(Q)

S 4a1||vllix+ 2b§T(Q)||v||i/

S (401+ 2bfT(Q))Ilvllir

g (4a1 + 2bi’T(Q))R2

 

 

Therefore, for V2) 6 V

2||v||v||A(v) - onl

 

v- + r(B(v)QB*(v) — BonBav) S wllvllt+(4a1 + 2537(9))35
This proves (2.17) with to satisfying (2.18), thus the assertion holds.

Example 2.1.1 Consider the following stochastic evolution equation:

du(t) = A0u(t)dt + F(u(t))dt + B(u(t))th (2.19)

with initial condition

u(0)=cp€H

Suppose A0, F and B satisfy the following conditions:

31

(i) A0 : V —+ V“ is coercive so that there exist constants a > O and A, for Vv E V,
2<v, on> S All'vlliz - OIllvlli/
(ii) F: H —» H and B: H —+ L(K,H) satisfy: forv E H.
||F(v)||fz + ||B(v)||i(x,u) S K(1+||v||i;)
(iii) For u,v E H,
”FM - F(v)||i1 + tr((B(U) - B(v))Q(B‘(U) - B‘(v))) S Allu - v”?!-

If the solution {u0(t),t Z 0} of du(t) = Aou(t)dt is exponentially stable( or even

exponentially ultimately bounded), and as ||v||H -+ 00

||F(v)||H = 0(ll'vllH), ||B(v)||L(k.H) = 0(llvlln)-

then the solution {u(t),t Z 0} of (2.1.9) is exponentially ultimately bounded in m. s.

s.
Proof: Let A(v) = on + F(v) for v E V. Since F(v) E H,

2<v, A(u)> + tr(B(v)QB‘(v))
= 2<v, on> + 2<v, F(v)> + tr(B(v)QB*(v))
= 2<v, on> + 2(v, F(v)) + tr(B(v)QB'(v))
S Allvllii — allvllf/ + 2llvllHllF(vlllH + ||B(v)|li(x,n)tr(Q)
S A'llvllf; - 0||v||f1+ 7
for some constants A’ and '7, hence equation (2.19) is coercive. Under additional

assumptions (ii), (iii), the strong solution {u(t),t Z 0} of (2.19) exists ([21], Th 3.1).

By assumption (ii)

||F(v)ll§1 + T(B(v)QB‘(v)) S ||F(v)llii + llB(v)|lL(K,H)T(Q)
_ (1+ 7(Q))K(1+ llvllit)

/\

32

and since

l|F(v)llH = 0(||v||H).T(B(v)QB’(v)) S llB(vlllL(K,H)T(Q) = 0(llvllh)

as ”UHH —> 00, the assertion follows from corollary 2.2.

Remark 2.1.3 the above example extends to infinite dimensions the corresponding

results in Zakai [2.9] and Miyahara [18]. As an application, we derive the following.

Example 2.1.2 Stochastic heat equation. Let S1 be the unit circle and W(-, ) a

Brownian sheet on [0, 00) x S". We consider the following stochastic heat equation:

8X(t) _ o5X(t) 91w
7,—(5) — 79?“) — axwo + 150(c)(5)) + b(X(t)(£)) 8,8,, (220)

with initial condition

X(0)(-) = $(-) 6 L2(51).

where a is a constant and f,b are real-valued functions.
Let
H = L731), v = WWS‘), Ao(:c) = (— — 0).,

and F and B given for 6 E S1 and x,y E L2(Sl) are defined by

Fort) = mu», Bum/1(5) = b($(€))y(€)-
Let

ll-‘vllH = (jslxzdefi forer

uxnv = (Lunggrwoi formev.
Then

2 < 55,1405 >= —2||$||fz + (-20 + 2)llilillh S -2||$||iq + (-2C5 + 2lll$lll¥ = JON-5‘”?!-

33

Therefore, by Theorem 2.1.1 and Remark 2.1.1 the solution of dx(t) = on(t)dt is
exponentially stable if a > 0. furthermore, if, in addition we assume f and b are both
Lipschitz continuous and bounded, then from Example 2.1.1, the solution of (2.20) is

exponentially ultimately bounded in m. s. 5..

Example 2.1.3 Consider the following SPDE:

2321i

dtu(t,x)= (02 8—71".

+ flg—- :+ 7.. + g( ))dt+(01% + 02U)dW(t)

with initial condition
u(0,x) = q5(x) E L2(—oo,oo)flL1(—oo,oo),

where W(t) is a one-dimensional standard Brownian Motion.

Let
H 2 L2(— —oo ,oo), V = H6(—oo,oo)
26211 (9a 8
A(u) = 012T+B5§+7u+g B(u)=ola—:+ogu

nun” = (f:u5dx)2 foruEH

llullv = (f:(u5 +(Z—:—)5)dx>

NIH

for u E V
Suppose g(x) E L2(—oo,oo)flL1(—oo,oo). For 2) E V.
2<v, A(v )> +tr(BvQB*v )
=2/_:(v T33W}:)+7v+gdun+/_:(o1—+or,.v)2da:
-—- (—2a5 + 015?“th + (27 + a: + 2a5 — emu”... +2 /_ w(v.g)dx
3 (—2a5 + canvnt + (27 + a; + 2a5 — a: + 6)l|v||i: + Eugut
for V6 > 0. Similarly for u, v E V,

2<u — v, A(u) — A(u)> + tr(B(u — v)QB"(u — v))

= (~202 + Uflllu - vllix + (27 + 03 + 202 - 03)”?! - vlli;

34

By ([21], Th. 3.1), if -2012 + 012 < 0, there exists a unique strong solution
u‘p(t) E L2(Q,C(0,T; H))flM2(0,T : V).

Now we want to find its Lyapunov function explicitly. Taking Fourier transform of

the SPDE:
dtfl(t, A) = (—02A2Tt(t, A) + iAflfi + 727(t, A) + §(A))dt
+(i01’\fi(ta A) + 0260. A))dW(t)
= ((-Ot2A2 + 2A3 + 7)fi(t, A) + §(A))dt

+0011 + 02)fi(t, A)dW(t)

Now for fixed A, let

a = —02A2 + iAB + 7
b = §(/\)
c = iolA + 02

By the result in the appendix,

— —A A a a c‘ - _
E[H(t,A)[2 = {E]&(A)[2 + 2Re(bb(:+b‘1;(+):a)'(+fi +':E)c))}e(a+a+cc)t
Haw) + b) e... "

—2Re( a(a + CE)

 

) + 2Re(

 

)

a(a + 21‘ + CE)
By the Plancheral theorem, with H = L2(—oo, oo)
||u“’(t, ')||i1 = ”WU, 'llllf
Hence
Ellu“"(t)lliz = Ewan}. = E l... |fi(t,A)l2dA
= /°° E|&(t,A)|2dA

6u¢(t)
8x

 

EIIu5<t>IIt = Ellu‘”(t)lli; + En Mi. = /_ 00(1 + A’)E|fi(t.z\)|2dA

35

For a suitable T > 0,
T t
11,.) = f / E11151 (s1115vdsdt

: [OT / f: (1+A2)E (t A)12dAdsdt

= f:(1+15)/T/E1a(tA)15dsdtdA

The above computation of the Lyapunov function is very complicated, but if we use
Corollary 2.3, we just need to compute the Lyapunov function of the linear SPDE,
which is much simpler. let {u0(t),t Z 0} be the solution of

M82

dtuuv’r): (02 6—15

2 + 3%— :+ 7u)dt + (01%— + UgU)dW(t )

then
d¢u0(t, A) = aiio(t, A)dt + ciio(t, A)dW(t)

a, c are defined as above. We can solve du(t, A) explicitly:
a0”, A) 2 510(0, A)e at—-c2t+cW(t) _ —<,0(/\)6 at—%c2t+cW(t)

Elton, A15 = 1¢1A115e<555+55>5

It is easy to see

+00 _ _
t—+ Ellu3(t)llt = / (1+ A511¢1A115e<5+5+55>5dx

is continuous for V90 6 V, and
”A(v) - Ao(v)||v- = llgllv- = 0(Hvllv) as Ilvllv —5 +00
r(B(v)QB'(v) — BonBgv) = 0

since B is linear. Therefore if {u0(t),t Z 0} is exponentially ultimately bounded in m.
s. s., the Lyapunov function Ao(<,o) of the linear system is also a Lyapunov function

of the nonlinear system , and for a suitable T > 0,

00 T t
110(10) = f oo(1+15) f0 f0 E1ao(t,1)15dsdtd,\

36

e{(—202+a¥)A2+27+a§}T
H402 + 0f)/\2 + 27 + 0;}?
T 1
_ (—2a5 + atW + 27 + at — {(—2a5 + 0W + 27 + 0312

 

= / (1+ A2)|15(/\)|2(

00
-oo

 

 

)dA

Therefore the solution of the nonlinear system SPDE is also exponentially ultimately

bounded in m. s. 3..

Remark 2.1.4 From the above computation we see, if we replace E]|u"’(s)|[¥, by

E]|u‘”(s)||§, in Ao(c,o), then the leading term of Ao(<,o) is

°° WM]2
T/_oo (2a2 — of)A2 — 27 -— 0% d)“

this does not satisfy the first inequality of(ii) of theorem 2.1.1.

 

2.2 Ultimate Boundedness and Invariant Measures
In the previous section, we considered ultimate boundedness:
lim sup E||u¢(t)||%, S M for V99 6 H (2.21)
t—++oo

for the solution of (1.1) and gave a sufficient condition for (2.21) in terms of a Lya-
punov function in Corollary 2.1.1. In this section will study the existence of invariant
measures for {u(t)} under ultimate boundedness. First we will see the result in
H = R”.

Let IR(x) denote the indicator function of the set{x E H, ||x||H > R}, with R > 0,
we have the following result, see ([13], PP 72):

Theorem 2.2.1 If H = R", and a Markovian semigroup (P1) is Feller, then an

invariant measure 11 for (P1) exists if and only if for some element x E H,

. . . 1 T
121—13100 li‘IEigTA PtIR(x)dt — O.

37

the key in the proof of the sufficient condition of this theorem is that {x E
H, lellH S R} is compact when H = R", but this fails to hold when H is a Hilbert
space. But if

V 5—» H

is compact, then {x : Hva S R} is compact in H, therefore, we can have the following

counterpart result in Hilbert spaces as in R".

Theorem 2.2.2 Let TR(x) denote the indicator function of the set {x E H, [[xllv >
R}, with R > 0. Suppose V H H is compact and a Markovian semigroup (Pt) is
Feller. Then a sufficient condition for an invariant measure p for (Pt) exists is there

exists some element 1,0 6 H, such that

Rum“ 11m1nf— / Pth(cp :0. (2.22)

T—’+OOT

On the other hand, if there exists an invariant measure [1 for (Pt) with support in V,

then {2.22) is also necessary.

For equation (1.1), the semigroup

I)=/Hf(y)( p(tn)dxy

is Markovian and Feller, therefore, we can apply the above theorem to the solutions

of (1.1) and get:

Theorem 2.2.3 Suppose V 5—» H is compact. Then a sufficient condition for an
invariant measure 11 for the solutions of {1.1) exists is there exists some element

Lp E H, such that

T
lim liminf-é—f P{[|u“’(t)[|v > R}dt = 0. (2.23)
o

R—r+oo T—o-l-oo

0n the other hand, if there exists an invariant measure [1 for the solutions of (1.1)

with support in V, then (2.23) is also necessary.

38

Now if we use the coercivity condition, we can get the following sufficient condition

for the existence of invariant measures of the solutions of (1.1):

Theorem 2.2.4 Suppose V <——+ H is compact, and the solution {u(t),t 2 0} of (1.1)
under coercivity condition (1.2) is ultimately bounded (in H - “H norm ). Then there

exists an invariant measure p for {u(t),t _>_ 0}

Proof: Applying Ito’s formula (1.4) to ”9.9”?“ taking expectation and applying coer-
civity condition (1.2), we get

t
ElluutMli—Iwu = t. Ecuuusmids

A tE ‘9 2at tE "’ 2d
[0 Nu (any s—a/O uu (any 8+7t

|/\

hence
t 1 t
[0 Euuusmidssgofo Ellu¢(3)||i1d3+||90||i1+7t),

therefore,
1 T
if Humvuv >R}dt
0
T 2
s Th Ellu‘°(2tfllvd,

(Ill/0R |___|90||H
<___ ‘p
OR,( Ellu (t)||Hdt+——— T +).

Since we assume {u"°(t),t Z 0} IS ultimately bounded, for fixed 4,00, there exist two

constants To and M, such that
E||u“’°(t)||%, g M for t 2 To
Therefore,

T
lim liminff—ll; P{||u“’°(t)||v > R}dt

R—»+oo T—>+oo

_ 900( 2
<REE‘mli‘i‘iEofanzThT EH“ )HH dt
"(fro EH” ¢°( t)||Hdt+/TT E||u¢°( (illlii dt)

w 1
< Elm. 13.123732 T

__ ‘Po 7-
(BRIBwliEigfanzr(/R EH“ 0“” dt+M(T m)
=0

39

Therefore the assertion of the theorem follows.

For the application of the above theorem, let us see the following example.

Example 2.2.1 (Stochastic Navier-Stokes Equation [26]) Let D Q R2 be a bounded

domain with smooth boundary 00. Consider the equation:

v.‘ ,1: 8—1.” ,x 21)- '3'
{1592+ +Zj= '1 vJa :_lm+V23=Iaan-+U’Wt(x)

x, p 8.1:.

'~?_ 8” =0, xED,i-—-1,2, u>0

J-1 8.1:,
Let (33° 2 {v E [C°°(D )]2: V - v = 0} (V- is gradient) and H the closure of C8° in
[L2(D)]2,V = {v 6 [H6(D )]2 : V - v = 0}. It is known [26] that

[L2(D)]2 = H ea Hi
Where H L is the orthogonal complement of H characterized by
i = {v = V(p), for some p E H1(D)}
Denote by H orthogonal projection from [L2(D)]2 to H i and define for v 6 (38°,
13(2)) = VHAv — H[(v - V)v]

Then B can be extended as a continuous operator on V to V‘, and V Q H E V" is
a Gelfand Triplet with V H H compact. The equation can be recast as a stochastic

evolution equation in the form:
du(t) = B(u(t))dt + adW(t)
u(O) = 6, 66 V a.e.
where W(t) is a H -valued Q-Brownian Motion. We observe ([26],PP. 347) that the

above equation has an unique strong solution {u€(t),t _>_ 0} satisfying:

 

Euua" )IIH+ E / Eu", “(t)IIHdt<EuiuH+ +-tr(Q)

i=1

Hence using the fact that ||u£(t)||v is equivalent to ( 3:1 Ila—g:i.9||§,)i , we get

liminfl fT E(nu‘u)“2 dt < —C—tr(Q)
T—ooo T 0 V _ 2V

40

where c is a constant. Using Chebychev’s inequality we get

1 T
o a o _ 5 —
Rl-{r-Poo ler‘I—I-obngl) P(”u (“NV > R)dt — 0

By the above remarks we get that invariant measure exists and the support of it is

in V.

The idea used in the above example is the relationship between ultimate bound-
edness of {u“’(t)} in H —norm and boundedness of % fOT E (||u"’(t)||%,dt in addition to

the compactness of embedding of V H H.

Remark 2.2.1 As a consequence we easily get a result on the existence of the in-
variant measure of the stochastic heat equation ([20], [25]). As we see in Example
2.2, the solution of the stochastic heat equation is ultimately bounded in m. s. 3., and
since V ‘—) H is compact by .S'obolev embedding theorem, the existence of a invariant

measure follows.
Example 2.2.2 We consider the equation of the form:

du(t) = —Au(t)dt + F(u(t))dt + B(u(t))dW(t), u(O) = (,0 E H,
where F, B satisfy the conditions in Example 2.1.1.

The above model with A = —A occurs in the work of Funaki [9] on the random
motion of string problem. Funaki gave an explicit form of the invariant measure
in the case B E 1. However since A is coercive [14], we get that the solution is
ultimately bounded in m. s. s.. In view of the fact that A has pure point spectrum
with eigenvalues Ak ~ ~k2, we get by [12] that it has an invariant measure.

Furthermore, we get conditions on the finiteness of the second moment of invariant

measures as in the following theorem.

Theorem 2.2.5 Suppose V ¢—+ H is compact, and the solution {u(t),t Z 0} of (1.1)

under coercivity condition (1.2) is ultimately bounded (in H - "H norm). Then any

41

invariant measure u of {u(t),t 2 0} satisfies

[v IITIIde) < oo

Proof: Let f(x) = Hxllfi, and fn(x) :2 X[0,,,](f(x)), where x is a characteristic function.
We note that fn(x) E L1(V, p), By the use of Ergodic theorem for Markov process
with invariant measure([28],PP. 388), there exists the limit

. 1 T
11m T/o Ptfn(x)dt = f;(x) (u — a.e.)

T—++oo

and
Eflf; : Eufna
where Eufn = fv fn(x)u(dx).

From the assumption of ultimate boundedness of {u“(t),t 2 0}, there exists a

positive constant M > 0, such that
lim sup E||ux(t)||§, S M for Vx E H.
t—r+oo

By the same argument as in the above theorem, we have

Imam—é: TWIIu )llvdt<1imSUP-(1; —' (—/ Ellu"’()ll"£zdt+|——|1;.$””+)SM’L|
T-v+oo 0
hence,
H 1. 1 T
fH(:v) = T133100%/0T Ptfn($ (T)dt<I¥T1+::°pf 0 Ptf($)dt
1 M__|_)I|
= r <
133813110 THE“ (t )Ilvdt_ 0
therefore, the fact that fn(x) T f(x) implies
Euf=n 11m Efn— 11m Ef <MlA——|

This proves the assertion.
This improves the result in [3] on the parabolic Ito equation and gives more
information about the invariant measure studied in ([25], [20]). The existence of

invariant measure for parabolic Ito equation can be proved using Corollary 2.1.1 with

Map) = ”90”}, and Theorem 2.2.4.

Chapter 3

Weak (Weakly Positive)

Recurrence of the Strong Solution

In this chapter, we will study the weak recurrence and weakly positive recurrence
properties to compact sets for the strong solution of the stochastic evolution equation
(1.1) under the the condition of ultimate boundedness. Using the results in Chapter

2, we study the problem in terms of Lyapunov functions.

3.1 Ultimate Boundedness and Weak Recurrence

In this section, we study weak (positive) recurrence of the solution of (1.1) to a
compact set under the condition that it is (exponentially) ultimately bounded in m.

S. 8..

Definition 3.1.1 A stochastic process X(t) defined on H is weakly recurrent to a

compact set if there exists a compact set C, such that
P1,.{w : X(t) E C for some t 2 0} =1 foer 6 H.

where PI stands for the conditional probability under the initial condition X(O) = x,

the set C is said to be a recurrent region.

42

43

Remark 3.1.1 Throughout this paper, weakly (positive) recurrent means weakly (pos-

itive) recurrent to a compact set, instead of to a bounded set as in [18, 12].

Theorem 3.1.1 Suppose V c_, H is compact, and the solution {u(t),t Z 0} of (1.1)
under coercivity condition {1.2) is ultimately bounded in m. s. 3.. Then {u(t),t Z 0}

is weakly recurrent.
This theorem is proved through a series of lemmas

Lemma 3.1.1 Let X(t) be a strong Markov process on H, if there exist a positive
Borel measurable function p(x) defined on H, a compact set C and a positive constant
(I, such that

Px{w : X(p(x)) E C} 2 6 > 0 for Va: 6 H. (3.1)

then the process X(t) is weakly recurrent and C is a recurrent region.
Proof: For fixed :1: E H, let

T1 = p(T).

91 1’ {WZX(T1) ¢ C},
72 = Tl +P(X(7'1))I

92 = {w i X(Tzl ¢ C},
7'3 = 72+P(X(72))H
0:3 = {W1 X(Tsl 63 C},

52,-.

'38:

12...:

1

Since {an : X(t,w) ¢ C for any t Z 0} Q (200, it is sufficient to show P4000) = 0. By

“
II

the assumption,

Px(nl)SI—6<lv

44

and since p : H --) R is Borel measurable, T,- is a stopping time for each i, by the use

of strong Markov property, we have

PH(Qlfl92) = Ex(E$(Xs.anlfn))
= Ex(Xa.Ex(Xn2|}-n))
= Exbm E(XnH|X(T1)))
= Er(X91PX(n){w X(P(X (Tl)))¢C})

By assumption (3.1),

PX(nI{w = X(p(X(T1))) 93 C} S 1— 5,

hence
P3621 flflg) S (1 — (5)2.

In the same manner,
Pr(flQ,-)S(l—6)"—+0 as n—>oo.
This proves P4000) = O and X (t) is weakly recurrent.

Lemma 3.1.2 Let X (t) be a continuous strong Markov process on H, if there exist
a positive Borel measurable function 7(x) defined on H, a closed set C and a positive

constant 5, such that

(x)+1
[7 Px{w:X(t) e C} 25>OfoerE H. (3.2)

7(3)

then there exists a positive Borel measurable function p(x) defined on H, such that
7(17) S P013) S 7($)+1, and

Px{w : X(p(x)) E C} 2 6>0. for Va: 6 H.

45

Proof: By (3.2), 7(3)“ Px{w : X(t) E C} 2 6 > 0 for Va: 6 H, hence there exists

W(r)

t, E [7(x),'y(x)+1), such that
Px{w : X(tx) E C} 2 5.

Define
p(xI = mm 6 [7(T),7(T)+1),Px{w=X(t) e C} .>. 6}.
Since the characteristic function of a closed set is upper semicontinuous and t —> X (t)
is continuous, t ——) Pr{w : X(t) E C} is upper semicontinuous for each fixed x E H,
therefore,
MW 3 X(P($)) E C} 2 5.
Now what we need to show is x —> pn(x) is Borel measurable.

For each t Z 0, define B¢(H) = 8(H). For any fixed T > 0, since X(t) is a
Markov process, the map (t,x) —I Px{w : X(t) E C} of [0, T] X H into (R1,'R1)
is B([0,T]) x B(H) measurable, hence it is B([0,T]) >< BT(H) measurable, therefore
(t,x) —I Px{w : X(t) E C} is a progressive process w. r. t. {Bt(H)}t20, by ([7], Cor.

1.6.12), .7: ——> pn(x) is Borel measurable. This proves the lemma

Let D, = {x : ||x||v S r} for any real number r and let D, be the closure of D, in

(H, I] - Hy), D: the interior of D, in (H, II - Hg), and Di 2 H— D,., then (D,)c = (Df.)°.

Lemma 3.1.3 Suppose the solution {u(t),t Z O} of {1.1) under coercivity condition
(1.2) is ultimately bounded in m. s. s., i.e., (2.21) holds. Let M1 = M +1, then there

exists a positive Borel measurable function p(cp) defined on H, such that
C O 1
Pcplw : “(“80” 6 (Dr) is E(IAlMi '1' M1 ‘1' '7) (3'3)
for any positive number r and any go 6 H.

Proof: Since lim sup E“’||u(t)||§, S M < M1 for ch E H, hence for each (,0 E H, there
t—ioo

exists a positive number To, such that

E‘p||u(t)||,2q 3 M1 for t Z Tw

46

Let
7(99) : inf{t : E‘pllu(s)]|§, S M1 for all s 2 t}.

Since t —) E“°||u(s)||§{ is continuous, E‘pllu(7(<p))||§, S M1, and

{so 2 W?) S t} = {so = E”IIU(8)||iI _<_ M1 for 61118 2 t}

= fl {¢=E¢||u(8)lli;SM1}-

Hanseo
and since «,0 ——> E“’||u(s)|]%, is Borel measurable, {so 2 7(Lp) S t} 6 B(H), therefore,
«,0 ——> 7(30) is Borel measurable.

Now we apply Ito’s formula (1.4) to IIxIIfH, take expectation and make the use of

coercivity condition (1.2), we get

EWIIUHW) + 1)l|12q - E¢IIU(7(90))||§I
W(SP 1
= f H E£llu(s)llizds

We)
u(w)+1 2 w(¢)+1 2
s A EEIIuIsIIIHds—a/ ErIIu(sIIIHds+~I
W(v) 7(90)

hence

7W)“ 9p 2 1 7W)“ «p 2 so 2
[M EllU(S)llvds s —(A [M E IIu<sIIIHds+E IIU(7(90))||H+|7|)

HQ

S -(|/\|M1 + M1 +|1|)

Q

Using Chebychev’s inequality we get

( )+1 ( )+1 r 2
f7 ‘p P,p{w : ||u(t)||v > r}dt S /:: El—Izzmv-dt
w w

1
< _
ar2(lAlMl + M1 + '7”:

use)

hence

w(w)+1 C O l
/ mu : u(t) e (0,.) }dt 3 E(IAIMI + M1+I7|),

W(sp)

therefore

1(¢)+1 _ 1
/ ma : u(t) e D.}dt21— W(WMl + M1 + m).

1(90)

47

By Lemma 2.2, there exists a positive Borel measurable function p(cp) defined on H,

such that
7(99) S p(so) S 7(99)+1, and
— 1
PH{w = u(p(<p)) 6 Dr} 21— Q(IAIM1+ M + 17]) for Vso e H. (34)
Therefore

C 0 1
av = u(p(so)) 6 (Dr) }S gin/w. + M1+ Ivl)
for any positive number r and any (,0 E H.

Proof of Theorem 3.1.1: From (3.4), we can choose r large enough such that:

P¢{w : u(p(cp)) 6 Dr} Z for V90 6 H.

Since V H H is compact, D, is a compact set in H, by Lemma 3.1.1, u(t) is weakly

recurrent to D...

now we consider weakly positive recurrence of the solution of (1.1) to a compact

set under the condition that it is exponentially ultimate boundedness in m. s. 8..

Definition 3.1.2 A stochastic process X (t) defined on H is weakly positive recurrent
to a compact set if there exists a compact set C, such that X (t) is weakly recurrent

to C and the first hitting time to C has finite expectation for any x = X (0) 6 H.

Theorem 3.1.2 Suppose V H H is compact, and the solution {u(t),t Z 0} of {1.1)
under coercivity condition (1.2) is exponentially ultimately bounded in m. s. 3.. Then

{u(t),t 2 0} is weakly positive recurrent.

Proof: Since the solution {u(t),t 2 0} of (1.1) is exponentially ultimately bounded
in m. s. s., we suppose (2.1) is satisfied. Let M1 = M + 1, then it is easy to see if

t > ln(1+ Cll‘Plliqla then E‘p||u(t)||}, S M1. Let

.1.
-I0

W(t): %ln(1+ clz),

48

then W(t) satisfies:
Evllufllllh S M1 for V90 6 H and t Z W(llcpllu),

and

 

2 2.2 < 00 for any N 2 0. (3.5)

Let

 

I, l
A = EVIAIMI + M1+|7l(1+ e),
E0 = 5]"
E1 = D—(l+l)K-—DIK = E(I+I)K “(Dildo for 121,
W’(l) = W(lKao)+1 (3.6)

where 010 is the constant such that “xHH S aollxllv for Vx E V. As in the proof of

Lemma 2.3, there exists a Borel measurable function p(cp) defined on H, such that

W(ll‘PllH) S 10(99) S W(ll‘PllHl'l‘l and
P¢{W1“(P(<P)l e (DfH-)°} s W(IAIMI + M1+|7|)
1
= W for cheH. (3.7)

Let

T1 = 10(17),

1131(0)) = “(Tliw)i

01 = {w = 351(0)) ¢ Eu},
7'2 = 71+ P($1(w)),
33200) = “(Tzawla

02 = {W1$2(w)¢ E0},
Ta = 72 + P($2(w)),

x3(w) = u(7'3,w),

49

Q3 2 {w : x3(w) ¢ E0},

9,.

:38:

(100:

'0.
ll
j—i

By the proof of Lemma 3.1.1, when 6 > 0, we know

P¢(fl St.) = 0
i=1
therefore,
9 = U Q? :2 U{w : x,-(w) E E0} a.e. (Hp).
'=0 '=O

Let

i—l i-l

A.- = 0? — U 9? = 950“) n.)
i=0 i=0
2 {w : x1(w) ¢ E0, - - - ,x,-_1(w) ¢ Eo,x.-(w) E E0}.

then

9 = Z A, a.e. (Hp).

i=0

For i Z 2, let’s further divide A,- as

A.-.—_ Z A.,I,,...,I,_,

ll i"'1li—1 21

where A;,11,...,1,_1 = {w : $10.0) E E11, - - - ,$;_1 E E1,_,,x,-(w) E E0}.
Let T(w) be the first hitting time to E0, then for w 6 A1 = (2‘13,

T(w) S p(cp) S W(Ilcplln) + 1,

for w E Ai,ll,°",li—17

7(a)) S 7,-(w) = Ti-1(w) + p(xi—1(w))-

Since when w E A:,1,,...,1,_,,

CC;_1(LU) 6 Eli—1 g Eat—1+1)!“

 

50

hence
||$i-1(W)||H S aollIEi—1(w)||v S ao(l.-_1 +1)K
then
Man-1(a)» S W(||(x,-_1(w)||H) +1 S W(ao(l.-..1 + 1)K) +1 = W’(l.-_1 +1)
then
T(w) S 77-1 + WIUi—l + 1).

Therefore by induction, for w E Ai,1,,...,1

i—l’
7(a)) S W(Ilcplly) + l + W'(li +1)+°°'+ W'(l.'—1 +1).
On the other hand, by the strong Markov property

P¢(A,-,1,,...,I,_,)

P.p{w : x1(w) 6 E11, - - - ,x;_1(w) E EI,_,,x.-(w) E E0}

P¢{w : x1(w) 6 E1,, - - - ,x.'_1(w) E El._1}

P¢({w I 113100) E Em ° ' wan—2(a)) E E1.-2ln{w I $i-1(w) E E,,_,})

E¢{X{w=$1(w)€E11.'-',r1—2(W)EEI,_2} ' PEI—2(a)){w’ : “(p(mi-2(w)awl) E Eli—1}}

|/\ |/\ ll

Since E1“, = D—(l._1+1)K fl(Df'_1K)°, by (3.7) we have

Pxi—2(W){w’ : u(p(x.-_2(w),w') E Eli—1}

S Pxi—2(W){wl : ”(p(xi-2(w)aw’) E (DT,_1K)O}
l
< _—
_ li—1(1+5)2’
hence
l
-. <————-P : E ---,.-_ E. ,
PIA/11.11. Ill—1l—li2_1(1+€)2 <P{w 310'") E ’11 (E 2(w) E 11—2}

by induction,

1 1

P‘P(Ai,l1,m,l.—1) — (1 + C)2('_1)l¥' . . 112—1 a

 

51

and

P,(A1) < 1.

Therefore,

E‘”[T]
s 23 PH(A1.1..-~,1.-_.)[W(||solln)+1+W’(ll+1)+---+W’(l.-_.+1)]

i711 i"'ili—l 21

°° 1
1 —————.——
W(llSOHH) + + Z; (1 + 6)2(:—1)

Z W(HSOHH)+ 1+ W’(11 + 1) + - - - + W’(l.-_1 +1)

|/\

 

l1,",li_121 IT...li2-l
°° 1
= (IIIHIIH +1+Z(l 61)., —-———1,{(W(llsolln)+l) Z -———, ,2_
i(=2 (1,-- -,.'l_.1>1 '- l
. W' l +1
+(z _1) 2 12(1):: )
11"".11—121 l i-l

= W(IIIHIIH +1+Z€f€1727m{(w(llrlln)+l)4"il+(-1)A"2B}

 

= (W(IIHIIHI+1I(1+:((1 f€)2I‘I+1I——;;—,B)z(—1f )2) )‘ 2(1—1)

where A = 22:, [l2 and B = 22:1 217W’(l + 1) which is convergent by (3.5) and (3.6).
Hence we see if we choose 6 large enough, E‘p[7'] is finite. Since V H H is compact,

E0 is compact, the assertion of the theorem holds.

3.2 Weak Recurrence and Lyapunov Functions

In chapter 2, we studied the relationship of ultimate boundedness and Lyapunov
functions. Combining Theorem 2.1.5, 2.1.6 and the corollaries there and Theorem
3.1.1, 3.1.2 here, we immediately get the following results, these results give conditions

in terms of Lyapunov function for weak and weakly positive recurrence.

Theorem 3.2.1 Suppose V H H is compact. Let {u(t),t 2 0} be the solution of
equation (1.1) satisfying coercivity condition {1.2). If there exists a function A : H —+

52

12 satisfying the following conditions:

(i) A satisfies {1.3),

(it) Cll|99||i1 - k1 S A(<19) S Callvlli; + ksifor V<P E H,
(iii) LAW) S —02A(80) + k21f07‘ V90 E VI

where c1(> 0),c2(> 0),03(> 0),k1,k2 and k3 are constants. Then {u(t),t Z 0} is

weakly positive recurrent.

Theorem 3.2.2 Let A satisfy (i), (iii) in Theorem 3.1 and
(121’ c.IIHIIH - k. s A(HI forvH e H

for some constants c1(> 0) and k1, then {u(t),t Z 0} is weakly recurrent.

The weakly positive recurrence of the solution of the nonlinear equation (1.1) can also
be studied through its first order approximation. Let {uo(t),t Z 0} be the solution
of the linear SPDE (2.11). We suppose that the linear operators A0, B0 satisfy the

coercivity condition (2.12) and the other conditions posted there.

Theorem 3.2.3 Suppose V H H is compact and the solution {uo(t),t Z 0} of the
linear equation (2.11) satisfying coercivity condition {2.12) is exponentially ultimately
bounded in m. s. 3.. Let {u(t),t 2 0} be the solution of the nonlinear equation (1.1).
Furthermore, we suppose A(v) — on 6 H for all v E V. Iffor v E V,

2||v||H||A(v) - onllH + T(B(v)QB‘(v) - BonBEI'v) S wllvlliq + k (33)

with w, k constants and

C

w < .
005% + 9.51% + £13} + g) + 71%|MQ+ $1 + gm

 

(3.9)

Then {u(t),t Z O} is weakly positive recurrent.

53

Corollary 3.2.1 Suppose V H H is compact and the solution {uo(t),t 2 0} of the
linear equation (2.11) satisfying coercivity condition (2.12) is exponentially ultimately
bounded in m. s. 3.. Let {u(t),t 2 0} be the solution of the nonlinear equation (1.1).
Furthermore, we suppose for v E V, A(v) — on E H, and

”A(v) — onllii + 7(B(’U)QB*(U) — BonBS'U) S K(1+||v||§,)
for some constant K > 0. Iffor v E V, as ||v||H —> oo
”A(v) — onHH = 0(IIUIIH) and T(B(U)QB'(U) — 307162350): 0(llvllirl-
Then {u(t),t 2 0} is weakly positive recurrent.

Theorem 3.2.4 Suppose V H H is compact and the solution {uo(t),t Z 0} of the
linear equation (2.11) satisfying coercivity condition (2.12) is exponentially ultimately

bounded in m. s. 3.. Furthermore, we suppose t —-> E||uo(t)||§, is continuous for all

so 6 V. Let {u(t),t 2 0} be the solution of the nonlinear equation (1.1). Iffor v E V,

2||v||v||A(v) - onl v- + 7(B(v)QB*(v) - BonBSU) S WIlvllb + k (3-10)

 

with w, k constants and

C

< 0
(ac +1I13[(§+ 91%)6 + $31 + g) + figflg + $11,} + 5)?)

 

(3.11)

Then {u(t),t Z 0} is weakly positive recurrent.

Corollary 3.2.2 Suppose V H H is compact and solution {uo(t),t Z 0} of the
linear equation (2.11) satisfying coercivity condition (2.12) is exponentially ultimately
bounded in m. s. 3.. Furthermore, we suppose t —-1 E||uo(t)||¥, is continuous for all
(,0 E V. Let {u(t),t 2 0} be the solution of the nonlinear equation (1.1). Iffor v E V,

as Hvllv -—1 oo
“A(v) — onllw = 0(II‘UIIV) and 7(B(’U)QB‘(U) - BonBSv) = 0(ll'vlli/l-

Then {u(t),t 2 0} is weakly positive recurrent.

54

3.3 Parabolic Ito Equations and Examples

Let D C R" be a bounded domain with smooth boundary 3D, r be a positive integer.
Let

V = W"2(D), H = W0'2(D).

By Sobolev imbedding theorem, V H H is compact.

 

 

Let
0"“ 3""
A (x = aa(x a , 3.12
0 ) lag,” )61211 BIBS” ( )
where a = (011, - - - ,an) is a multiindex and la] = al + - - - + an.

Garding’s inequality ([22], Th. 7.2.2) says that if A0 is a strongly elliptic operator,

then it is coercive.

Example 3.3.1 Consider the parabolic Ito equation of the form:

du(t,x) = Aou(t,x)dt + f(u(t,x))dt + B(u(t,x))dW(t)
u(0,x) = (p E H (3.13)

ulap = 0

where A0,f and B satisfy the following conditions:
(i) A0 : V ——> V“ is a strongly elliptic operator
(ii) f: H —+ H and B: H —1 L2(K,H) satisfy: forv E H.
“f(vlllii + ||B(v)||i.(x,n) S K(1+||v||i1)
(iii) For u,v E H,

||f(U) - f(vlllfi + tr((B(u) - B(v))Q(B*(U) - B‘(U))) S /\||u - “0“}?-

If the solution of equation du(t,x) = Aou(t,x)dt is exponentially ultimately bounded

in m. s. s., and as HvIIH —> oo

||f(v)||H = 0(||v||H), ||B(v)||L(k.HI = 0(||v||H)I

55

then the solution {u(t),t 2 0} of (3.13) is exponentially ultimately bounded in m. s.

s.(example 2.1.1), hence it is weakly positive recurrent.

Example 3.3.2 Consider the following 1-dimensional parabolic Ito equation:

du(t)“: (0123—3 +55§ + 7'“ +9($ ))dt + (013—: + UzuldWU)
u(0,x) = (15(1) 6 L2(D)fl L1(D) (3.14)

ulap = 0
where D = [0,1] (_3 R1, W(t) is a 1-dimensional standard Brownian Motion. Let
V = W1’2(D), H = W0’2(D).
Suppose g E L2(D)fl L1(D). Take A(x) = ||x||il for x 6 H, for v E V,
£A(v) = 2<v, A(v )> + tr(BvQB‘v )
3
= 2/D(v ,a ”+3 +1+g)+/D(al—-+azv)2da:
= (~2a +af)llvllv+ (21w: +2a —a§IIIvIIH+2/ (v,gIdx
S (_ _220 +allllvllv+(27+02+202 ‘01‘1' E)llvllHi' +‘llgllil

for V6 > 0. Hence if —2c12 + 012 < 0, then the coercivity condition (1.2) satisfied.

Furthermore,

I
EA(v I< (— —2a + a.III— a IIH+ (21 + a: + c)||vl|§1 + gllgllir,
by Poincare Lemma, Ila—ill}! > 8”va thus,
1
£A(vI 5 (—16a2 + 8a? + 21 + at + eIIIvII'H + gllgllt,

Therefore if
—16012 + 80? + 27 + 0.3, < 0

then Theorem 3.2.1 says that the solution {u(t),t Z 0} of (3.14) is weakly positive

recurrent.

 

Chapter 4

Stability and Ultimate
Boundedness of the Mild Solution

In this chapter we will study the stability, exponentially ultimate boundedness and
stability in probability for the the mild solution of the stochastic semilinear evolution
equation. The main technique is to construct an appropriate Lyapunov function.
Once this is done, we will exploit the methods developed in chapter 2 of this disser-

tation and those in [15] to obtain results for the mild solution.

4.1 Exponential Stability in the Mean Square Sense

The exponential stability in the mean square sense of the mild solution of (1.5) was
undertaken in a systematic manner in [10, 2, 6], and was continued in [15] for the
strong solution under coercivity condition. An example was given in [15] to show that
the usual Lyapunov function was not bounded below. In this section, we construct a
new Lyapunov function and show that the existence of such a Lyapunov function is
a necessary and sufficient condition for the mild solution of (1.5) to be exponentially
stable in the m. s. s.. Then we use this bounded below Lyapunov function to study

the problem of the stability in probability, we conclude that exponential stability in

56

57

the m. s. 3. implies stability in probability for the mild solution of the semilinear
evolution equation (1.5).
Let us assume that u"’(t) is the mild solution of (1.5), we say it is exponentially

stable in the m. s. s. if there exist positive constants c, B, such that
E]|u“’(t)||2 S ce'fltllcpllz. for all (p 6 Hand t > 0. (4.1)

The next theorem gives a sufficient condition for u“’ (t) to be exponentially stable

in the m. s. s., it was proved in [11], we quote it here for the ease of reference.

Theorem 4.1.1 The mild solution u‘p(t) of (1.5) is exponentially stable in the m. s.
s. if there exists a function CE(H) 3 A : H —-+ R satisfying the following conditions:

(i) clllsoll2 s A(HI s €3||Tl|2a (42)
(ii) £A(HI s —cHA(HI, (4.3)

for V90 6 H, where c1,c2,c3 are positive constants.

Proof: Apply Ito’s formula (1.7) to ec?‘A((p) and un(t) and take expectation, where
un(t) is the strong solution of (1.8), then

eE‘EA(u::(tII — A(u:(0II = E /,t W + EHIA(u:(sIIds
By (ii),

C2A($0) + CHAW) S -£A(s0) + £HA(<.0)
= < A'(HI. (E(nI — 1)F(99)> +§1r(A"(HI(R(nIE(HIQ(R(nIE(HII* — B(<P)QB*(<P)))

therefore,
erEMuutII — A(u:(0II
E [0 e°2’(< A'(u:(sII, (E(nI — IIE(u:(sII >

+ étr(A”(u‘.f(s))(R(n)B(u:(s))Q(R(n)B(u:';(s)))* — B(H:(s))o3*(u:(s)))))ds.

|/\

58

Let n —+ 00, by the dominated convergence theorem and Theorem 1.5.3, we get
GCQ‘EMWUD S A(v),
hence by(i), we have:
CIEHUW)“2 S EMU‘pUll S E’Q‘MIP) S Esta—”tllrllz-

This proves the theorem.

Now we want to construct a Lyapunov function if the solution u“’(t) of (1.5) is
exponentially stable in the m. s. 3..

First, let us consider the following linear case. Suppose F E 0 and B = Bo is

linear, Then equation (1.5) has the following form:

{ du = Audt + BoudW(t) (4 4)

u(O) = (,0.
We assume ||Box|| S d||x|| for Vx E H and the solution of this equation is 115(1). The

infinitesimal generator £0 corresponding to this equation is £0 A(go) =< A’ (4p), Acp >
+%tr(A”(<p)BosOQ(BosO)*)-

Theorem 4.1.2 If the solution 113 (t) of equation (4.4) is exponentially stable in the
m. s. s., then there exits a function A0 E CE(H) satisfying (4.2) and (4.3) with L
replaced by £0.

Proof: Let

A.(HI = )0” EIIuatIIrdt + aIIHIr (4.5)

where 01 is a constant to be determined later. Since 113(1) is exponentially stable
in the m. s. 3., [6” E [lug (t)||2dt is well defined and there exists a symmetric and

nonnegative operator R E L(H) [6], such that

(0 EIIuatIIrdt =< 1290,90 >

59

and
£0 < Rear >= -|l<p||2o
Hence,
40(99) =< 1390,99 > +allrllz- (4-6)

It is obvious that A0 E C§(H) and c1||(.o||2 S A0((p) S (||R|| +oz)||<p||2, this proves (4.2).
To prove (4.3) with .C replaced byllo, we note that A is the infinitesimal generator of
a Co-semigroup S(t) satisfying “S(t)” S e‘“, there exists a constant A, without loss
of generality, we assume it is positive, such that < (p, A(p >S /\||(,o||2 ([12]), hence we

have,

£o|l<pll2 = 2 < WE > +tT(Bo<ITQ(BosO)‘) S (2) + d2t7‘(Q))||99||2- (4-7)

Hence,

50400?) = 130 < 399190 > ‘I'CY£0||4P“2

-|Irl|2 + 0(2) + d2tr(Q))ll(p||2
S (-1+ 0(2) + d2t7‘(Q)))||<P||2- (4-8)

l/\

Therefore, if a is small enough, (4.3) holds with I. replaced by£o. This proves the

theorem.

Remark 4.1.1 In [10], Haussmann proved a stability theorem under conditions H1 :
3c > 0,7 > 0 such that ”S(t)“ < ce"7‘ for Vt > O and H2 : ”ft;>0 S{A(I)Stdt|] < 1,
where < A(I)go,1/) >= tr(B((p)QB"(w)).

Define Ao((p) = f0°° E||u3’(t)||2dt + a||(p||2, This is well defined because of H1 and

H2. From our theorem, use Ao(cp) as a Lyapunov function, the result follows.

For the nonlinear equation (1.5), to assure zero is a solution, we need to assume
F(0) = 0, 8(0) = 0. If the solution u“’(t) is exponentially stable in the m. s. 8., we

can still construct a Lyapunov as in (4.5):

A(HI = [0 EIIuruIIrdt + aIIHIr

60

But it may not be in C:(H). If we assume it is in C52(H), we claim that it satisfy
(4.2) and (4.3). Now, let us prove this claim.

Since u‘*°(t) is exponentially stable in the m. s. s., we assume it satisfies(4.1), hence
f0°° E|]u‘P(t)]|2dt S fillcpllz for all x E H, therefore a||(p||2 S Ao(go) S (% +a)||(p||2, this
proves (4.2).

To prove (4.3), let

WI = [f EIIu:(tIII2d1.

Observe

EW(u‘”(r)) = E /°° E(llu“”"’(s)ll"lu‘”(r))ds
0
But by the Markov property of the solution of (1.5), this equals
[0 E(E(||u““°"’(s)||2lf2‘))ds
where £3” = o{u‘*0(r), T S r}. The uniqueness of the solution implies

E(lltt"¢(')(8)ll2lff) = E(llu‘pts + MUSE")-

 

Hence
E‘Il(u“’(r)) = /°° EHu‘p(r + s)I|2dH = /°° E||u“’(s)||2ds. (4.9)
O r
By the continuity oft —1 E||u“0(t)||2, we get:
d
mm = ;;(E‘I’(U"’(T)))|.=o
ip _
= lim E‘P(u (rII WI
r—>0 r
_ ' 1 r (p 2
— [1337/0 EIIu (sIII ds
= -||<19||2
Therefore,
EMT) = NO?) + 05”er

-||<P||2 + 0(2 < EIAE + FM > +tT(B((P)Q(B(90))‘)
S -||<19||2 + Qalllrllz + 0(2 < 3. FOP) > +tr(B(90)Q(B(90))’)-

61

Since we assume F(O) = 0, 3(0) = 0, using the Lipschitz condition (1.6), we get
We) 3 -ll<ell2 + 11(2) + 2d + d2tr(Q))l|90||2-

Hence if a is small enough, A((p) satisfies (4.3). Therefore, we have proved the

following theorem:

Theorem 4.1.3 If the solution u‘P(t) of (1.5) is exponentially stable in the m. s. s.,
furthermore, F(0) = 0,B(0) = 0, and \Il((p) = f0°° E||u‘*’(t)||2dt is in CE(H). Then
the function A((p) constructed above satisfies (4.2) and (4.3).

As in the case of strong solution, we also have difficulty to show W((p) E CE(H). We
thus turn to use the first order approximation to study the exponential stability in

the m. s. s. of the solution of the nonlinear equation (1.5).

Theorem 4.1.4 Suppose the solution 113(1) of the equation (4.4) is exponentially
stable in the m. s. s., and it satisfies (4.1). Then the solution u“’(t) of (1.5) is

exponentially stable in the m. s. 3. if

2||99||||F(99)ll + T(B(99)Q(B(<p))* - BoIOQ(Bos0)‘) < gllrll2 (410)

Proof: Let Ao((p) =< Its/2,1,0 > +a||(p]|2 as defined in (4.6). Since 113(1) satisfies (4.1),
”R“ S 5. Since Ao((p) E CE(H) and satisfies (4.2), if we can show that Ao((,o) satisfies
(4.3), then by using Theorem 4.1.1, we are done. Since

EAH(HI — Evo(HI
< A((HI. E(HI > +%tT(/\3(90)(B(<P)Q(B(<P))* — EHHQ(EHHI*II

2 < (R + cum, F (4.0) > +tr((R + a)(B(s0)Q(B(<p))" - BorQ(Bo<P)‘))
2(IIRII + a)l|<pllllF(so)|| + (IIRII + a)T(B(<P)Q(B(90))’ — EoHQ(EoHI*I
(IIRII + a)(2l|so||||F(r)l| + r(E(HIQ(E(IoII* — EHHQ(EHHI*II. (4.11)

|/\ || ||

here we used lemma 2.1.2. By (4.8) and the assumption (4.10), £Ao((p) satisfies (4.3)

if we choose a small enough. This proves the theorem.

62

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

below.

Example 4.1.1 Consider the following SPDE:

32' 8
dtu(t, x) = (020—; + Bag + 7u)dt + audW(t)

with initial condition

u(0,x) = (f)(x) E L2(—00,00)flL1(—00,00).

where W(t) is a one-dimensional standard Brownian blotion.

Let
8%. Bu
_ 2 _ _ 2_ _
H — L( 00,00), A(u) a ax2+fl6$+7u,
8(a) = cu. IIuII = (/ uzdeT.

Now we compute E||u¢(t)||2 explicitly. Taking Fourier transform of the SPDE:

wan, A) = (—a2/\217(t, A) + 115.7(1)) + 712(1, 1))111 + 017(t, A)dW(t)
= (—a2/\2 +1Ap + ma, A)dt + ail(t, A)dW(t).

Solving the equation we get for each fixed /\:
E|fi(t,»\)l2 = l$(/\)|26“2“2*2+21+°2".
By Plancheral theorem, with H = L2(—00, 00)
llud’UI')“2 = “17%, ')||2

we have

Ellu"(t)||2 = EIIar(tIII2=E/°°Ia(t,AII2dA

= [00 [$(A)[28(—2012A2+27+02)td)\

-OO

63

If we assume 27 + 02 < 0, then we get
E||u¢(t)||2 S ||¢||26(27+”2)‘,

hence the solution of the SPDE is stable. But,

A 1
I000 Ellum ”2'2“” I: |¢(A)|2202A2—(27+02)d)‘

thus the usual Lyapunov function ffooo Elluf’(t)||2dt is not bounded below.

 

4.2 Stability in Probability

Stability in probability is studied in [15] for the strong solution under coercivity
condition. The key in [15] is the construction of a bounded below Lyapunov function.
In this chapter, we will use the bounded below Lyapunov function constructed in the
previous section to study the problem of the stability in probability. We conclude
that the exponentially stable in the m. s. 3. implies the stability in probability for
the mild solution of (1.5). We first present a general result following [15].

Theorem 4.2.1 Let u“’(t) be the solution of equation (1.5). If there exists afunction

A((p) e C§(H) having the following properties:
{1'} AW) -* 0 as ”99H —* 0.
(ii) infllgp||>e A((p) = AE > 0,
(iii) £A((p) S 0 when ”(pH < 6 for some small 6.
Then

“liHmOP{sup [Iu‘p (t )|| > e} = Ofor eache > 0

i.e., zero solution of equation (1.5) is stable in probability.

64

Proof: We first obtain the inequality

P{sup ||u"”(t)|| > e} S -A—/(\f—)-for(p E H.
t

To prove this, let CC = {x E H : Ila/2]] < e},'rc = inf{t : ||u“’(t)|| > 6}. Using the same

technical as in Theorem 4.1.1 and condition (i), (ii), we get
A((o) 2 EA(u‘p(t /\ 71))2 A¢P(7',E < t).

this proves the inequality. Now let x —+ 0, we get the assertion.
The function constructed in Theorem 2.2 for the linear equation (4.4) satisfies the

conditions of Theorem 2.5, hence we get the following theorem.

Theorem 4.2.2 The solution 113(1) of the linear equation (4.4) is stable in probability

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

For the stability in probability of the zero solution of the nonlinear equation (1.5),

we have the following theorem.

Theorem 4.2.3 If the solution u§(t) of the linear equation (4.4) is exponentially

stable in the m. s. s., and

2llrl|||F(r)ll + T(B(<p)Q(B(<P))‘ - BosoQ(BosO)') < wllcpllz (4-12)

for (.12 small enough in a sufl‘iciently small neighborhood of (p = 0. Then the zero

solution of the nonlinear equation (1.5) is stable in probability.

Proof: Since the solution 113(1) of the linear equation (4.4) is exponentially stable
in the m. s. s., we define Ao((,o) =< R90,(p > +oz||cp||2 as in (4.6). By (4.11) and

assumption(4.12), we get

£Ao(‘10) S 0-

Obviously, Ao(go) satisfies the other conditions of Theorem 2.4, therefore our assertion

holds.

65

4.3 Exponentially Ultimate Boundedness in the Mean
Square Sense

Exponentially ultimate boundedness in the m. s. s was studied by Wonham [27], Za-
kai [29] and Miyahara [18] in terms of a Lyapunov function for the finite dimensional
case, and Miyahara constructed a Lyapunov function if the solution of the stochastic
differential equation is exponentially ultimately bounded. Ichikawa [12] gave a suf-
ficient condition for the mild solution of a semilinear stochastic evolution equation
to be exponentially ultimately bounded in terms of a Lyapunov function. In chapter
2 of this dissertation, we studied the same problem for the strong solution of SPDE
under coercivity condition, and get a necessary and sufficient condition in terms of a
Lyapunov function for the linear case and use the first order approximation to study
the nonlinear case. In this section, we study this problem for the mild solution of (1.5)
and also give a necessary and sufficient condition in terms of a Lyapunov function for
the linear case and use the first order approximation to study the nonlinear case.
For exponential ultimately boundedness in the m. s. s. we have a similar result

as Theorem 4.1.1 for exponential stability in the m. s. 3..

Theorem 4.3.1 The mild solution u‘p(t) of (1.5) is exponentially ultimately bounded
in the m. s. 3. if there exists a function Cb2(H) 9 A : H —+ R satisfying the following

conditions:
(i) c1||(,0||2—k1 S 4(0) S CBHIOH2 ——k3, (4-13)
(ii) £460) S -621\(<0)+k2. (4-14)

for V30 E H, where c1(> 0),c2(> 0),c3(> 0),k1,k2 and k3 are constants.

Proof: The proof of this theorem is similar to that of Theorem 4.1.1.
For the converse problem, we first see the linear equation (4.4). We have the

following theorem.

66

Theorem 4.3.2 If the solution 113(1) of equation (4.4 ) is exponentially ultimately
bounded in the m. s. s., then there exits a function A0 E CE(H) satisfying (4.13) and
(4.14) with L replaced by Lo.

Proof: Suppose the solution 113(1) of (4.4) is exponentially ultimately bounded in m.

s. s., i.e., we suppose (2.1) holds. Let

A.(HI = foTEllut(s)l|2d-s + aIIHII': (4.15)

where T is a positive constant to be determined later.

First Let us show A0 E CE(H). let

T
110(HI = )0 EIIuatIIPE.
Using (2.1), T
_ C
1110(1)»: )0 (ce mllrllz + M)dt s Ellrllz + MT. (4-16)

If [[90]]? = 1, then
110(0) 3 §+ MT.

Since 113(1) is linear in x, for any positive constant k, we have
I:
“090“) = (Wat)
hence,

\Po(k<p) = k2‘1’0(90)

therefore, for any x E H

17
llrll

Let c’ 2 5+ MT then \Ilo((,0) S c’]|(p||2 for V90 E H.

10(HI = H99H2‘I’of I s (g + MT)||<0||2-

Let
T .1
That) = f E < u:(sI,u.(sI > ds for w e H
0

67

then T is a bilinear form on H, and by using Schwartz inequality, we get
T '1!
IT(H,iII = I [0 E < u1(sI,u.(sI > dsl

T , 7 1
f0 (Ellut(s)I|2)5(Ellu6”(s)||2)2ds

T , T ,_
()0 Ellut(s)l|2ds)5(/O EIIut(sIII*dsI2
= \I'oMWoWI
C'HSOH ' Hit)“-

|/\ |/\
up.-

|/'\

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

7(9011b) : (0901 11b) (417)
and
“C” = SUP |(C<P,¢)| S 6'.
||10||=1I||¢||=1

Since @060) = 700,42) = (090190)) so
113(0) 2 2C(p and ‘11"(90) 2 2C

Hence, \Ilo E CZ(H) and A((p) E CE(H).
By (4.16) and the fact that Ao((0) Z allapllz, (4.13) satisfies.

By the same reason as we get (4.9), we have
T 2 T+r 2
ETo(ut(rII = /o El|u6°(r+s)ll ds = / EIIu:(sIII ds.

By the continuity oft —+ E||ug(t)]|2, we get:

£09100?) = (Ti-.(E‘pduarllllmo
= lim Ew°(“g(r))‘E‘1’0(<P)

_ - 1 r .p 2 _1_ ”T (a 2
— hm(—;]o EIIu.(sIIIds+,,/T EIIu.(sIIIdsI

r-oO

~l|¢||2 + EllU3(T)||2
-||<0||2 + CE’MIIITIIZ + M
(-1+ ce-BT)||(,H||2 + M

 

|/\ ||

|/\

68
Thus, using (4.7),

£0A0(so) = £o‘1’o(s0) + afiollsoll2
3 (-1+ ce‘”)llsoll" + am + dztr(Q))HsoII2 + M- (418)

Therefore, if T > l—"B—C, then we can choose a small enough such that A0(go) satisfies(4.l4)
with 5 replaced by £0.

Consider the solution of the nonlinear equation (1.5). If it is exponentially ulti-
mately bounded in m. s. s., using ideas similar to the stability problem, we can still
construct the Lyapunov function as A(go) = fOT E||u¢(s)||2ds + a||go||2, but it may not
be in CE(H). But if it is in CE(H), follow the proof of this theorem and Theorem 2.3,

we can show it satisfies (4.13) and (4.14). Therefore, we have the following theorem.

Theorem 4.3.3 If the solution u“‘°(t) of (1.5) is exponentially ultimately bounded in
the m. s. s., and W(cp) = If Ellu‘p(t)||2dt is in C52(H) for some big T > 0, then there
exits a Lyapunov function for u‘p(t) satisfies (4.13) and (4.14).

Now we use the first order approximation to study the properties of exponentially
ultimate boundedness in the m. s. s. of the solution of the nonlinear equation based
on the same property of the solution of the linear equation. As in Theorem 2.5, we

have the following theorem.

Theorem 4.3.4 Suppose the solution u: (t) of the equation (4.4) is exponentially
ultimately bounded in the m. s. s., and it satisfies {2.1). Then the solution u“’(t) of

(1.5) is exponentially ultimately bounded in the m. s. 3. if

2||<19||||1”’(<p)|| + T(B(90)Q(B(so))' - Bo¢Q(BosO)‘) < w||90||2 + M1 (419)

for any constant M1 and

_ ‘38
w < max 1;. (4.20)
a)? E "l" M3

69

Proof: Let Ao(cp) be the Lyapunov function as defined in (4.15) with T > l—“fi—C such
that (4.20) gets its maximum at T. We just need to show that Ao(go) satisfies (4.14).
Since Ao(<p) = (Cgo,<p + a||cp||2 for some 0 E L(H, H) with ”C” 5 5+ MT and

a very small. Following (4.11) we have

5AM?) — £vo(<p)

S (HCH + a)(2ll<p||||F(99)ll + T(B(90)Q(B(<p))* - Bo¢Q(Bos0)*))
s (g + MT + among)“: + M1).
Using (4.18),
£AO($0)

s (—1 + ce’fiTHIsoIP + a(2A + d2tr(Q))lls0||2 + M

H; + MT + a)(WI|90||2 + M.)

= (—1 + ce"flT + co(E

[3

+a(2/\ + d2tr(Q) + w)||<pll2 + M + (3 + MT + a)M1
[3

Since (.0 satisfies (4.20), —1 + ce‘fiT +w(% + MT) < 0, we can choose a small enough
such that (4.14) is satisfied.

+ MTllll99||2

Corollary 4.3.1 Suppose the solution ug(t) of equation {4.4) is exponentially ulti-

mately bounded in the m. s. 3.. If as “so” —> oo

||F(<P)|| = 0(ll99ll) and 7(B(90)QB*(<P) - BosOQ(Bo<P)") = 0(||<P||2)

then the solution u“’(t) of (1.5) is exponentially ultimately bounded in the m. s. s

Proof: Since as ||<p|| -—> oo

||F(<P)|| = 0(Ils0||) and T(B(<p)QB*(90) - Bo<pQ(Bo<p)") = 0(||<P||2)

For any fixed w satisfying (4.20), there exists an K > 0, such that

QHSOHHFW)” + T(B(<P)QB’(¢) - Bo¢Q(Bo<p)*) S wllwllz

70

for all ||<,o|| _>_ K.
But for Hap” S K, by the Lipschitz condition,

2||¢|H|F(¢)Il + T(B(90)QB*(90) - Bo¢Q(Botp)‘)

S ll<p||2 + ”PM”2 + T(B(<p)QB*(<p)) + T(BosOQ(Bos0)’)
S llwll2 + ”FM“2 + (||B(<P)||2 + llBo(sO)||2)tr(Q)

s “99”? + 2d2(1+ Ilsoll2)(1+ MG»)

3 K2 + 2d2(1+ K2)(1+ 2tr(Q)).

Therefore,

lewll||F(90)||+T(B(90)QB‘(90)-Bo<pQ(Bosv)*) S WH99||2+RZ+2d2(1+K2)(1+2t7‘(Q))-

The assertion follows from Theorem 3.4.

Chapter 5

Appendix

Computation of E|X(t)|2 of the following stochastic differential equation:
dX(t) = (aX(t) + b)dt + cX(t)dW(t)

where a,b and c are complex constants, W(t) is a standard one-dimensional real
Brownian Motion.

If b = 0, then X (t) can be computed explicitly:
X(t) : X(0)eat—%c2t+cW(t).
So we suppose b 75 0. Since

dX(t) = (aX(t)+b)dt+cX(t)dW(t)

dX(t) = (aX(t)+b)dt+EX(t)dW(t)

 

Where 7 means the complex conjugate. By Ito’s formula, we get

|X(t)|2 = X(t)X(t)
(0) + fotxswmi- [f)(—(Me) + fo‘d<x,x>.

><

= X(O)
= |X(0)l2 + (a + 5+ 66)]; |X(s)|2d3 + b/OtX(3)ds + b/otmds
(c + a) f; |X(s)l2dW(s)

71

72

Let «p(t) = E|X(t)|2, take expectation to the above equation:
t _ t t
p(t) = p(0) + (a + a + CE) [0 90(5)ds + b / EX(s)ds + b / EX(s)ds
o 0

Lemma: If 1%? = ay(t) + g(t), then

t
y(t) = game“ + / ecu-09W.

Proof: Proof is elementary.

Using this lemma, we have

_ _ t _ _ _
«p(t) = a(0)6<°+°+cc>t+ / e‘“+“+“)("’)(bEX(s)+bEX(s))ds
0

_ _ t _ _ _
= 90(0)e(“+“+cclt+2Re/ e(°+a+cc)(‘"’)bEX(s)ds

0

Now we compute EX(t), since
dX(t) = (aX(t) + b)dt + cX(t)dW(t)

X(t) = X(0) + /Ot(aX(s) + b)dt + 0/0: X(s)dW(s)

hence

EX(t) = EX(O) + a jot EX(s)dt + bt

Using the above lemma we get

 

 

b
EX(t) = —; + (EX(O) + ;)e°‘
Thus
99(t) : cp(0)e(°+a+cat + 2Re{— b3 (e(°+a+cat — 1)
a(a + a + CE)
E(EX(O) + .3.) (a+E+cE)t at
+ a + CE (e — e )}
therefore

 

bi + bEX(O)(a + a + ca) )e‘°+W"
(a + a + ca)(a + ca)

b(aEX(0) + b) 0, b5
a(Zi + c6) e )+ 2Rea(a + E + cE)

E|X(t)|2 = (E|X(0)|2+2Re

 

 

—2Re(

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