Luận án tiến sĩ: Stochastic delay equations and invariant measure for the wave equation with noise

First we study the moment stability of the trivial solution of a linear differential delay equation in the presence of additive and multiplicative white noise.. The stability of the firs

measure for the wave equation with noise

by

Xi Zhao

Submitted in Partial Fulfillment

of the Requirements for the Degree Doctor of Philosophy

Supervised by Professor Carl Mueller

Department of Mathematics

The College Arts and Sciences

University of Rochester Rochester, New York

2006

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Curriculum Vitae

The author was born in Shijiazhuang, Hebei province, China on December 24th,

1979 She attended the Department of Mathematics at the Nankai University, Tianjin, China in 1998 and graduated with a bachelor of science degree in Math- ematics in 2002 In the same year, she was admitted to the Department of Math- ematics at the University of Rochester, Rochester NY After earning a master

of science degree in 2004, she chose probability as her research area and was su- pervised by Prof Mueller as a Ph.D candidate She was granted a Teaching Assistantship from the Department of Mathematics during the academic years 2002-2006

Acknowledgments

I would like to express my deepest gratitude to my advisor Prof.Carl Mueller for his guidance, encouragement, inspiration, supervision and patience in the past four and half years It is him who introduced me to the area of probability and stochastic process and led me to stochastic differential equations and delay equations I am deeply impressed by his wisdom in thinking and solving problems

in probability, differential equations, analysis and physics I am fortunate to have him as my thesis advisor and greatly appreciate his time and efforts for each meeting, discussion, reading and correcting my manuscripts

I gratefully acknowledge those who served as my committee members in my oral defense: Prof C Mueller, Prof R Lavine, Prof D Gioev I thank them for very useful suggestions and discussions

I thank all the professors who taught me one or two courses during my Ph.D study, who pass their knowledge without reservation and teach us not only how

to learn but also how to think: Prof C Mueller, Prof S$ Segal, Prof R Lavine, Prof J Naomi, Prof J Harper, Prof A Greenleaf and Prof M Gage

Many thanks to Dr Kijung Lee, who led me through the reading course for two semesters I appreciate his thorough discussion for every important concept and quick reply to every question

I am very grateful to the secretaries of the Department of Mathematics: Joan Robinson, Fran Crawford, Hazel McKnight They cordially, faithfully and

patiently helped me with all administrative paper work since the first day I came

to the department

I am gratefully indebted to my family My parents have always been sup- portive and encouraging My dear husband Li’s love always strengthens, comforts and encourages me He takes most of the responsibilities to take care of our son when this thesis was being written

I have had the good chance to have many friends at University of Rochester who help to turn 4 years of study into 4 years of life I look forward to having fun with you again

This thesis is divided into two major parts First we study the moment stability

of the trivial solution of a linear differential delay equation in the presence of additive and multiplicative white noise The stability of the first moment for the solutions of a linear differential delay equation under stochastic perturbation is identical to that of the unperturbed system However, the stability of the second moment is altered by the perturbation We obtain, using Laplace transform tech- niques, necessary and sufficient conditions for the second moment to be bounded Then we establish the stability criteria for stochastic differential equations with Markovian switching using the comparison principle These criteria include sta- bility in probability, asymptotic stability in probability, stability in the pth mean, asymptotic stability in the pth mean and the pth moment exponential stability

of such equations Next, we study the uniqueness of the invariant measure for the wave equation with noise We will use a coupling technique and others from the theory of Markov chains on general state spaces The application of these Markov chain results leads to straightforward proofs of ergodicity of SDEs The key points which need to be verified are the existence of a Lyapunov function including returns to a compact set, a uniformly reachable point from within that set and some smoothness of the probability densities

3 Invariant measure for the wave equation with noise

3.1 Introduction: Coupling method 3.2 Ergodicity for the Markov chain through coupling 3.3 Application to the stochastic wave equation

ii iii

29

31

32

34 40

A Proof of Theorem 1.5

Bibliography

62

64

delay equations

Stochastic differential delay equations were first introduced by Ité [2] in the 1960s Fundamental results including existence and uniqueness of solutions, stochastic stability, numerical approximation, etc have only been developed in the last decade ([4], [7], [12], [13], [14], [16]) See [3] for a recent survey of these results

In spite of the efforts of many researchers, this field is still in its infancy For example, conditions for the moment stability of some linear stochastic differential delay equations with constant coefficients are still not known

The Lyapunov function method is useful to study the stability of differential equation and has been developed for both differential delay equations and stochas- tic differential equations In the 1990s, Mao extended this method to stochastic functional equations [7, section 5] Because of the results of Mao, we have some results for the stability of stochastic differential delay equations (see [7 Sec 5.6] for details)

In this chapter, we study the moment stability of the trivial solution of a linear differential delay equation in the presence of additive and multiplicative white noise The stability of the first moment for the solutions of a linear dif- ferential delay equation under stochastic perturbation is identical to that of the unperturbed system However, the stability of the second moment is altered by the perturbation We obtain, using Laplace transform techniques, necessary and

matical preliminaries for the linear differential delay equations needed for the rest

of the paper Then we will examine the effect of stochastic perturbations on the behavior of the mean and variance of the stochastic differential delay equation

Consider a stochastic differential delay equation of the form

dy = f(y, yr)dt + g(y, yr)dW (t) (1.1)

where y,(t) = y(t —7),7 > 0 and W(t) is a standard Wiener process

In the deterministic case

For any y*, such that f(y*, y*) = 0, y* is a steady state of (1.2)

Now linearize equation (1.1) around y*, we get

dz = (ax + bz;)dt + (ơạz + ơiz; + ơa)dW (t) (1.3)

where x(t) = y(t) — y* and a,b, o; are constants given by

Thus, we will study the equation

dz = (ax + bz,)dt + (oor + 0121 + o2)dW(t) (1.4)

tion

When o; = 0 in (1.4) , we have the linear differential delay equation

z'(t) = ax(t) + br(t — 1) (1.5) This differential equation has been studied extensively in [1] Now we will state

some of the main results

Definition 1.1 The characteristic equation of a homogeneous linear differen- tial equation with constant coefficients is obtained from the equation by looking

Lemma 1.3 (Existence and convolution of Laplace Transform) If f : [Ũ, oo) —

R is measurable and satisfies

\f(x)| sae" — t € [0, 00)

exists and is an analytic function of A for ReA > 6 If the function f * g is defined

by f « 9(t) = fo f(t — s)9(s)ds, then

L(f *9) = L(F)L(g)

The fundamental solution of (1.5) can be introduced in two equivalent ways It

is the solution of (1.5) whose Laplace transform is h~'(X) and equivalently, a solution of (1.5) with initial condition

+(86) = 0_ i_-1<6<0

1 if@=0

In what follows, we will denote by X(t) the fundamental solution of (1.5)

It is not hard to show that X(t) is bounded and |X(t)} < me” for some

constants m and n

Theorem 1.4 The solution X(t) of equation (1.5) with initial data given

above is the fundamental solution; that is

Also, for anyc >n

X(t) =Ƒ e*h“(A)dA — t>0

c

where n is the exponent associated with the bound on X(t)

see the proof in ref [1]

Now let C({—1, 0], R) be the family of continuous functions y from [—1, 0] to

R with the norm ||¿(6)|| = sup_i<e<o |¥(9)| Through the fundamental solution, the general solution of (1.5) with initial condition 7(0) = y(0), where —1 < 6 < 0,

is given by

ap(t) = X(t)e(0) + [ : X(t — 1 -s)y(s)ds (1.7)

Theorem 1.5 If ao = max{Re(X) : h(A) = 0} is negative, then for any

ag < a < 0, there is a constant K = K(qa), such that the fundamental solution X(t) satisfies the inequality

[X(t)| << Ke* (>0)

See the proof in the appendix

From Theorem 1.5, the solution of (1.5) with y(@) approaches 0 as t — oo if and only if ao < 0 The region in the (a, b)-plane such that ag < 0 is given in [1]

S = {(a,b) € R?| -asec€ <b<a,é =atané,a < 1,€ € (0,7)} (1.8) Now, we will give the estimation of œo and K(ø)

Let 4 = a9 +77 be the solution of the characteristic equation with the largest real part Substituting into (1.6), we have |a9 — al < |ble~°° , where ao is given

by the maximum real solution of

— a12 (a — a)? — b2e~22o + [areeos Ti =0 Lemma 1.6 For any @ > ao, the constant K = K(a) in Theorem 1.5 is given by

I (Aas 2m L b|e~% g(œ + /z)dz| + 7 Dyes z(z — | b|e~* )4z

_ (la — aole* + | d]) log 2

and inequality (1.9) follows

Note that (1.9) is defined when b # 0 and a > 0 When b = 0, we can simply take K(a) = 1 whenever a > do

The inequality (1.9) gives an estimate for K(a) for all parameters When

a <0 and |b| < —a, we have the following compact estimation for all bounds on Z6)

puting a» and K(a) with a > a in Theorem 1.5

bation

We now turn to a study of the system with noise, i.e., the parameters o; in (1.4) are not all zero Throughout this section, we will consider the Ité interpretation for the noise

From the fundamental solution X(t) in the previous section, the solution of (1.4) with the initial function

z(9) = (6), when -1<6<0 (1.13)

is a stochastic process in the probability space (Q, 4, P) given by

a(t) = ag(t)+ [ X(t s)(oox(s;¢) + ovm(si) + 02)4W(s)

Var[z(t)]

P(|x(t) — Ex(t)| > k) <

investigating the solution behavior and will be studied in this section We first define p-th moment exponential stability and p-th moment boundness

1.3.1 First Moment stability

Definition 1.8 The solution of equation (1.4) is said to be first moment expo- nentially stable if there is a pair of positive constants À and C’ such that

|EzŒ;@)| < Cllelle" — (t2 0)

for all € C({—1,0], #)

Theorem 1.9 If a9 = max{Re(A) : h(A) = 0} is negative, then for any ap <a <

0, there is a constant K, = K,(a), such that the mean of the solution

|EzŒ;2)| < Kilelle”“ @>0)

Therefore, if œọ < 0, then (1.4) is first moment exponentially stable

Proof: Taking expectation of both sides of (1.4), we have

Thus, we obtain a differential delay equation for the first moment Ex(t) By Theorem 1.5, we have Ex(t) — 0 as t — oo if and only if the parameter ap defined in Theorem 1.5 is less than 0, which proves the theorem

In fact, by (1.14) and the properties of Ité integral, we have

Ex(t) = X(t)o(0) + [ ec —1— s)£(s)ds (1.16)

1.3.2 Second moment stability

Definition 1.8 The solution of equation (1.4) is said to be p-th moment expo- nentially stable if there is a pair of positive constants \ and C’ such that

E(\z(t;¢) — E(@a(tsy))P) s Clly|Pe™* (20)

for all y € C((-1,0], 2) Furthermore, it is said to be p-th moment bounded if

there exists a constant A such that

E(zŒ;w) - E@œ(;2))f)<A (#20)

for all y € C([—1,0], R) Otherwise, the p-th moment is said to be unbounded

In the following, we will study the boundness of the second moment Hereafter,

we denote z(t;y) simply by x(t) We now turn to the behavior of the second moment of the solution x(t) From Theorem 1.9, the stability condition of the first moment is identical to that of the uninterrupted system and is determined exclusively by a and b Thus the stability of the first moment is independent of the parameters o; However, the situation of second moment is more complicated and depends on o; When o2 # 0, we can not expect the second moment to be exponentially stable Let A/(t) be the variance of the solution at time t, then the Chebyshev inequality yields

M(t)

for any k > 0 Thus, when the second moment is bounded, the solution of (1.4) is also bounded in some sense We will ask in this section when the second moment

is bounded for all t > 0 The following notation will be used

Let z(t) be a solution of (1.4), and define

M(t) = E(&(t)?), Mi(t)= M(t-1), N(t) = E((5( ~ 9), (1.19)

/ X2(t — s)(opBa(s;y) + øvEzi(s;@) + øa)2ds = F(t)

the third part is zero since E(t) = 0

Thus, equation (1.21) is established

To get the second moment stability, we will discuss in two cases:

Case 1: o9 = 0, = 0

When oo = o; = 0, we have the additive noise case and the second moment is given explicitly by

t

M(t) = 03 [ X(t — s)ds (1.23)

By Theorem 1.5, we have following result in the case of additive noise

Theorem 1.11 Let ap = max{Re(A) : h(A) = 0} H ơạ = ơi = 0, the second moment of (1.4) is bounded if and only if ay < 0 Furthermore, for any ap < a < 0, there is a constant K = K(a) such that

o3K?

2a

M(t) ~o? ir X?(s)ds < _ #2" cai (1.24) and the estimation of the second moment M(t) when t — œ is given by

When op # 0 or o; # 0, the noise at time t depends on z at time t and time (t—1) In this general case, there is no simple form for the second moment First,

where 5(À;) = 0 and ¢; are constants, fulfill our requirement

Therefore, we have the following necessary condition for the boundness of the

0<F(t)<(A+ Lời X*(t— s)ds < SO (1 — er"),

where A = A(a,y) is given above Thus, (1.29) is satisfied with

(A + | o2|)?K?

Let X(t) = X(t— 1) , then the functions X°(t) and X(t)X,(t) have Laplace transforms We will prove that the functions M(t) and N(t) also have the Laplace

transforms

The following theorem establishes the conditions for second moment of the solu- tion of (1.4) to be bounded

< Ka(1 ~ 99) + K*(|øa| + |oil)? [ M(s)ds

Here we have used (1.29), (1.31) and the fact that X(t) = M(t) = 0 when

—1<t<0 From the Gronwall inequality we have

M(t) < Koel (leolt orl)

and thus (1.34) is proved

Inequality (1.35) is a consequence of (1.34) and (1.36)

From Lemma 1.15, the Laplace transforms of M(t) and N(t) exist for all s > 2vy Lemma 1.16 There exists constants A2, v2, A3, v3 such that

Proof: From (1.21), we have

emo < see + |ogM(t) + 0?M,(t) + 2090, N(t)|

Inequality (1.38) can be derived in a similar fashion via a tedious computation,

the details of which we will omit

Lemma 1.17 If f(t) is differentiable on [0,00) , and

for constants A and v, then there exists vo > v and 0 < Lp < Ly; such that

In< z [ “am ƒ()ải <1¿ (141)

for all z € C with Re(z) > uo

Proof: From (1.40), we have

LF) SIAL + [> elf lat = Ly

and the lemma is proved

With these results, we can now turn to the proof of Theorem 1.14

Proof of theorem 1.14: We will divide the proof into several steps

Consider the equation for X?(t),

2

~ = 2aX? + 2bXX, Taking the Laplace transform of both sides, we have

—1+s£(X?) = 2a£(X?) + 2b£(XX\) therefore

1 2) —

“%)= s — 2a — 2bf(s) Thus, by (1.42) we have

(2) Let % = {max Re(s) : H(s) = 0} We will prove that for any 6 > Gp, there is

a constant Ky = K4(@) such that

To start, we will show that đọ < oo is well defined To do this, we prove that there exists A, and As such that if Re(s) is large enough

|f(s)| < Age and | g(s)| < Ase) (1.45)

We only prove the second inequality since the first one can be proved in an identical

fashion.

Let s = ut+iv with u sufficiently large and Lo, L; be the constants given in Lemma 1.17 By the Holder inequality and Schwarz’s inequality, we have

Y(t)= | Hr!(s)e°ds,

(e) where c is some sufficiently large real number and J) is defined as the following

1 c+iT

[y= Pal (e) - T>œ 271 Jc—¿T

At first, we have for any @ > đà,

Y(t) = (=f H-%) H-(s)e%ds Consider the contour integration of the function H~!(s)e*% around the rectangle e-iT >c+1iT — 6+iT — 6 -iT >c-— iT Then the integral is zero since (3) has no zero inside the rectangle Next, we only need to show that

Next, following the exact steps in the proof of lemma 1.6 (with h(A) being replaced

by H(s)), there exists K4 = K4(8) such that

If By < 0, we choose đọ < Ø < 0 and Ky as above Then

t

|M(t)| < [ (o2 + Kse%*-)) Kye"*ds

and thus the second moment M(t) is bounded for any initial function y(0) In - this situation, let

Moy = 02 / “Y()ds,

so

LA Kyoke® + pe — e%1), Thus, there exists a positive constant Xe = Xa(œ, Ø, @) such that

| M(t) _ M,.| < Ket maz(a,8)

ie., M(t) approaches M, exponentially when t — oo

If Øạ > 0, by the inverse Laplace transform, we have

Y(t) = O(e**) when ¢ is large enough We can choose an initial function y(@) such that

Ex(t) = Dycje**

when h(A;) = 0 and c; are constants Since Re(A;) < ao < 0, we have either

G(t) = O(t) as too when o2 #0 or

G(t) = O(e?") as too

for some a < ap < 0 when oz = 0 In either case,

Mụ) = Í ˆ@Œ — s)Y (s)ds = O(e®£)

when t — oo, and hence the second moment is unbounded The proof is complete Theorem 1.14 has established a criterion for the second moment of the linear stochastic delay differential equation to be bounded However, this criterion is not particularly useful The function g(s) in (1.32) involves the Laplace transform of M(t) and N(t) that are unknown In many applications, perturbations for system parameters only affect the right hand side of the equation that involves either the current state or the retarded state, and thus either o9 = 0 or o, = O In this situation, the function H(s) reads

H(s) = s — (2a+ 02) — 2bƒ(s) — ơ?e"*

and is determined by the system coefficients and f(s) that depends on the Laplace transform of X?(t) and X(t)Xi(t) Nevertheless, it is not trivial to obtain the explicit form of f(s) for a given system In the rest of this section, we will develop some estimations for f(s) and g(s) and present direct criteria for the second moment stability

Theorem 1.18 If b < 0, a0, < 0 and the equation

Ho(s) = s — (2a + 02) — 2bƒ(s) — ơ?e"*

has a non-negative solution, the second moment is unbounded

Proof: For s € R, we have

| * eX (E)X(t — Lat f(s) = ET < eo 9/2

for any s € R Thus, if s* is an non-negative solution of Ho(s) = 0, then H(s*) < 0 But H(s) > 0 when s is large enough and therefore the equation H(s) = 0 has a non-negative solution Thus, the theorem follows from Theorem 1.14

Corollary 1.19 If 6 < 0, o90, < 0, and either

satisfies 0 <u <1 and

—2log u — (2a + 02) — (2b + 2øgøi)}u — ơ?u? < 0 (1.47)

then the second moment is unbounded

Proof: When (1.46) is satisfied, we have Ho(0) < 0 When (1.47) is satisfied with 0 < u < 1, let s* = —2logu > 0, so Ho(s*) has local minimum at s* and Ho(s*) < 0 In either case, the equation Ho(s) = 0 has a non-negative solution and therefore the second moment is unbounded by Theorem 1.18

Theorem 1.18 and Corollary 1.19 tell us when the second moment is unbounded The following result tell us when the second moment is bounded

Theorem 1.20 If exists a <0 and K = K(qa) such that

Lemma 1.21 If y(t) is a non-negative continuous function on [—1, 00), and there are positive constants p and q such that

v(1) < p 9(S)đs +4 [ y(s — 1)ds + r(Ð (1.50) then for any Ø > 0 such that

8—p-qe>0

and

sup |r(£)e"#!| < œ t>0

there exists A = A(@), such that

S* = ——————| Ba page ‘sup ir(t)e sup ir(t) |

We will show that

Assume the contrary Let to be the minimum positive number such that S(to) = S*, then S’(to) > 0, while

Sa) < (p— B)S(to) + ge" Sto — 1) + (p + g)r(te™

< (p— B)S* + qeS* + (p+ q) -sup |r(t)e-™| t2

y(t) < R(t) +r(t) < |S(t) + r(the |e < Ae®

The lemma is proved

Proof of theorem 1.20: We have shown in the proof of Lemma 1.15 that

t

M@) < (|øo| + joul) f X?(t — s)((lo0|M(s) + |or|Mi(s))ds + F(t) (1.52)

From Theorem 1.5 and Lemma 1.13, we have ay < a < 0 and K = K(a), kK, = K2(a, yp) such that

X(t) < Ke’, OK< F(t) < K2(1-e?*)

Thus from (1.52), it follows that

Then

y(t) <p [ 9(5)4s +4 Í i(s)4s + r()

The inequality (1.49) implies

—9œ — p— qe?#“ >0 Thus by Lemma 1.21, there is a constant A such that

M(t)e*" = y(t) < Ae",

ie M(t) < A for all t > 0

From Theorem 1.14, if the second moment M(t) is bounded, it exponentially approaches a constant M, Let t — oo in (1.52) and apply (1.28), so we have

for any a and K (a) given in Theorem 1.20

From Theorem 1.9 and 1.20, for any parameter pair (a,b) in the region S, the first moment of the solution of the stochastic differential delay equation (1.4) approaches 0 as t — oo Furthermore, there exists P(a,b) > 0 such that if

([øo| + | oil)? < P(a,6)

the second moment of the solution is bounded and the upper bound is given by (1.53) as t > oo

From the estimation of K’(a) given in section 2, the function P(a, b) can be defined

as following

Let Po(a, b) = supagias)<aco “XI: For |b| < —a, define

P,(a,b) = —2(a + H(a, )),

where = /(ø, b) € (—| b|, —a) satisfies

ue?”" — |b|=0

We can define

max{Fq(a,b), Pi(a,b)}, — if0< |b|< —a

2 Stability criteria for

SDDE with Markovian

switching

Stochastic modelling has come to play an important role in many branches of sci- ence and industry An area of particular interest has been the automatic control of stochastic systems, with consequent emphasis being placed on the analysis of sta-

bility in stochastic models, and we here mention Arnold [18] (1972), Has’minskii [20](1981), Kolmanovskii and Myshkis [23] (1992), Kolmanovskii and Nosov [21] (1986), Ladde and Lakshmikantham [22] (1980), Mao [5] (1991), [6] (1994) and Mohammed [16] (1986) There has been little work on the stability of stochastic

differential delay equations with Markovian switching, although there are several papers on the stability of stochastic differential equations with Markovian switch- ing For example, Mariton [24](1990) investigated the stability of a jump linear equation

where r(t) is a Markov chain taking values in a finite state space; Basak [17](1996) has studied the stability of a semi-linear stochastic differential equation with Markovian switching of the form

dz(t) = A(r(t))x(t)dt + g(x(t), r(t))\dW(t); (2.2)