# Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps

Jianhai Bao , Aubrey Truman , Chenggui Yuan

## Abstract

The existence, uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented. The principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones. As a consequence, stability results of Basak et al. (Basak et al. 1999 J. Math. Anal. Appl. 202, 604–622) and Yuan et al. (Yuan et al. 2003 Syst. Control Lett. 50, 195–207) are generalized to cover a class of much more general stochastic partial differential delay equations with jumps in infinite dimensions. In contrast to the almost sure exponential stability in Ichikawa (Ichikawa 1982 J. Math. Anal. Appl. 90, 12–44) and Luo & Liu (Luo & Liu 2008 Stoch. Proc. Appl. 118, 864–895) and the moment exponential stability in Luo & Liu, we present a new result on the stability in distribution of mild solutions. Finally, an example is given to demonstrate the applicability of our work.

## 1. Introduction

Recently stochastic partial differential equations in a separable Hilbert space have been studied by many authors, and various results on the existence, uniqueness, stability, invariant measures, and other quantitative and qualitative properties of solutions have been established (e.g. Ichikawa 1982; Da Prato & Zabczyk 1992; Caraballo & Liu 1999; Liu & Truman 2002; Govindan 2003; Liu 2004; Luo & Liu 2008 and references therein). In particular, many scholars have paid much attention to stochastic partial differential equations with delays. By introducing the proper approximating strong solution system, several criteria for the asymptotic exponential stability of a class of Hilbert space-valued, non-autonomous stochastic evolution equations with variable delays were presented (Liu & Truman 2002). Caraballo & Liu (1999) and Govindan (2003) gave sufficient conditions for exponential stability in the p-th mean of mild solutions to stochastic partial differential equations with variable delays by using the properties of stochastic convolution and the comparison principle, respectively. Taniguchi (2007) and Wan & Duan (2007) considered by means of the energy equality exponential stability of energy solutions to non-autonomous stochastic partial differential equations with finite memory.

By contrast, there has not been very much study of stochastic partial differential equations driven by jump processes, while these have begun to gain attention recently. Röckner & Zhang (2007) showed by successive approximations the existence, uniqueness and large deviation principle of stochastic evolution equations with jumps. In Dong (2008), the global existence and uniqueness of the strong, weak and mild solutions to one-dimensional Burgers equation in [0,1], with a random perturbation of the body forces in the form of Poisson and Brownian motion, were studied. Under some circumstances, Luo & Liu (2008) established the existence of strong solutions to stochastic partial functional differential equations with Markovian switching and Poisson jumps, meanwhile the moment exponential stability and almost sure stability of mild solutions were investigated by the Razumikhin–Lyapunov type function argument and the comparison principle, respectively.

Most of the previous papers are concerned with the stability of the trivial solution either in probability or moment (i.e. the solution will tend to zero in probability or in moment). However, in many practical situations, such stability is sometimes too strong. Therefore, we want to know whether or not the probability distribution of the mild solutions to stochastic partial differential delay equations with jumps will converge weakly to some distribution (but not necessarily to zero). Such convergence is called the stability in distribution and the limit distribution is known as a stationary distribution. For the finite-dimensional case, Basak et al. (1999) made a first attempt to study the stability in distribution of a random diffusion with linear drift. By the Lyapunov function methods, Yuan & Mao (2003) and Yuan et al. (2003) generalized the results from Basak et al. (1999) to cover a class of more general stochastic differential equations and stochastic differential delay equations with Markovian switching, respectively.

As we know, the mild solutions do not have stochastic differentials. A significant consequence of this fact is that we cannot employ Itô formulae for mild solutions directly in most of our arguments. Therefore, the approaches of Yuan et al. (2003) are not available to deal with the stability in distribution of mild solutions to stochastic partial differential delay equations with jumps in infinite dimensions. Moreover, Luo & Liu (2008) studied the moment exponential stability and almost sure exponential stability of trivial solutions, which is different from our present topic–stability in distribution. The key contribution of this work is that by introducing an approximating system and constructing a suitable metric between the transition probability functions of mild solutions, we derive some sufficient conditions for the stability in distribution of mild solutions to stochastic partial differential delay equations with jumps. In consequence, we generalize some results of Basak et al. (1999) and Yuan et al. (2003) to infinite-dimensional cases.

The format of this work is as follows. In §2, we recall some preliminary results. Sufficient conditions for existence and uniqueness of mild solutions are presented in §3. In §4, several lemmas, which lay the foundation for our stability analysis, are presented. In particular, by introducing a suitable metric between the transition probability functions of mild solutions, the main results are derived. Finally, an example is given to demonstrate the applicability of our work.

## 2. Preliminary results

Let {Ω,,} be a complete probability space equipped with some filtration {t}t≥0 satisfying the usual conditions (i.e. it is right continuous and 0 contains all -null sets). Let H, K be two real separable Hilbert spaces and we denote by 〈.,.〉H, 〈.,.〉K their inner products and by ‖.‖H, ‖.‖K their norms, respectively. We denote by (K,H) the set of all linear bounded operators from K into H, which is equipped with the usual operator norm ‖.‖. In this work, we always use the same symbol ‖.‖ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. Let τ>0 and DD([−τ,0];H) denote the family of all right-continuous functions with left-hand limits φ from [−τ,0] to H. The space D([−τ,0];H) is assumed to be equipped with the norm . denotes the family of all almost surely bounded, 0-measurable, D([−τ,0];H)-valued random variables. For all t≥0, is regarded as a D([−τ,0];H)-valued stochastic process.

With the symbol {W(t),t≥0}, we denote a K-valued {t}t≥0-Wiener process defined on the probability space {Ω,,} with covariance operator Q, i.e.where Q is a positive, self-adjoint, trace class operator on K. In particular, we call such {W(t),t≥0} a K-valued Q-Wiener process relative to {t}t≥0. According to Da Prato & Zabczyk (1992, p. 87), W(t) is defined bywhere βn(t)(n=1, 2, 3, …) is a sequence of real-valued standard Brownian motions mutually independent on the probability space {Ω,,}, (λn,nN) are the eigenvalues of Q and (en,nN) are the corresponding eigenvectors. That is

In order to define stochastic integrals with respect to the Q-Wiener process W(t), we introduce the subspace of K, which endowed with the inner product,is a Hilbert space. Let denote the space of all Hilbert–Schmidt operators from K0 into H. It turns out to be a separable Hilbert space, equipped with the normfor any . Clearly, for any bounded operators , this norm reduces to . Let be a predictable, t-adapted process such thatThen, we can define the H-valued stochastic integralwhich is a continuous square-integrable martingale. For that construction, see Da Prato & Zabczyk (1992, pp. 90–96).

Let be a stationary t-Poisson point process with characteristic measure λ. Denote by N(dt,du) the Poisson counting measure associated with p, i.e. with measurable set , which denotes the Borel σ-field of K−{0}. Let be the compensated Poisson measure that is independent of W(t). Denote by the space of all predictable mappings for whichWe may then define the H-valued stochastic integralwhich is a centred square-integrable martingale. The reader can refer to Protter (2004) for the construction of this kind of integral.

In this work, we investigate stochastic partial differential delay equations with jumps in the following form: for given τ>0 and arbitrary t≥0,(2.1)with initial datum , −τt≤0.

Throughout this paper, for the existence and uniqueness of the mild solution to equation (2.1), we shall impose the following assumptions.

1. A, generally unbounded, is the infinitesimal generator of a C0-semigroup T(t),t≥0, of contraction.

2. The mappings F:H×HH, and are Borel measurable and satisfy the following Lipschitz continuity condition and linear growth condition for some constant k>0 and arbitrary x, y, x1, x2, y1, y2H,and

3. There exists a number L0>0 such that for arbitrary x, y, x1, x2, y1, y2∈Hand

For convenience of the reader, we recall two kinds of solutions to equation (2.1) as follows (see Luo & Liu 2008).

Definition 2.1

A stochastic process {X(t),t∈[0,T]}, 0≤T<∞, is called a strong solution of equation (2.1) if

1. X(t) is adapted to t and has càdlàg path on t≥0 almost surely and

2. on [0,TΩ with almost surely and for all t∈[0,T]

for any .

In general, this concept is rather strong and a weaker one described below is more appropriate for practical purposes.

Definition 2.2

A stochastic process {X(t),t∈[0,T]}, 0≤T<∞, is called a mild solution of equation (2.1) if

1. X(t) is adapted to t, t≥0 and has càdlàg path on t≥0 almost surely and

2. for arbitrary t∈[0,T], and almost surely

for any .

Remark 2.3

Let {X(t),t∈[0,T]} be a strong solution of equation (2.1), then by Luo & Liu (2008, proposition 2.1), we know that it is also a mild solution.

For our purpose, we introduce the Itô formula that will play a key role for our stability analysis as follows. Let R be real number, R+ be non-negative real number and C2(H;R+) denote the space of all real-valued non-negative functions V on H with properties:

1. V(x) is twice (Fréchet) differentiable in x and

2. Vx(x) and Vxx(x) are both continuous in H and (H)=(H,H), respectively.

Theorem 2.4 (Luo & Liu 2008, Itô formula)

Suppose VC2(H;R+), let be a strong solution of equation (2.1), then with t≥0,where , the domain of operator A,

## 3. Existence and uniqueness

In this section, we shall investigate the existence and uniqueness of the mild solutions to equation (2.1). In their book, Da Prato & Zabczyk (1992) have given a comprehensive description of stochastic equations in infinite dimensions without delay and jumps, the existence and uniqueness for both strong and mild solutions are given in the second part of their book; for stochastic partial functional differential equations without jumps. Caraballo et al. (2000) studied the existence and uniqueness of strong solutions and the result concerning the mild solutions was presented by Taniguchi et al. (2002); for stochastic evolution equations with jumps without delay, Röckner & Zhang (2007) have recently investigated the existence and uniqueness of strong solutions. In this section, to make our paper self-contained, we shall extend previous results to stochastic partial differential equations with delay and jumps, and discuss the existence and uniqueness of mild solutions to equation (2.1).

Theorem 3.1

Under the assumptions (H1), (H2) and (H3), equation (2.1) admits a unique mild solution.

Proof

We shall prove the existence and uniqueness of mild solutions to equation (2.1) by the contraction mapping theorem. Denote by 2 equipped with the normthe Banach space of all -adapted processes Y(t, ω):[−τ,T]→H, which are almost surely right-continuous functions with left-hand limits in t for fixed ωΩ. Moreover, Y(t, ω)=ξ(t) for t∈[−τ, 0]. For any t∈[0,T] and Y2, define the following mapping:To begin with, we show maps 2 into 2. In view of , we hence obtain by the assumption (H1) thatHowever, using the Hölder inequality, we compute from (H2)While, by virtue of Liu (2004, theorem 1.2.6), we obtain that there exists a constant C1>0 such thatIn addition, from Luo & Liu (2008, lemma 2.2) and (H2) as well as (H3), there is a positive constant C2 satisfyingIn consequence, maps 2 into 2. Next, we need to show is a contraction mapping. For any Y1,Y22, then(3.1)On one hand, by the Hölder inequality and assumption (H2), it thus follows that(3.2)On the other hand, taking into account (Liu 2004, theorem 1.2.6) and assumption (H2), we have for some C3>0(3.3)Moreover, from Luo & Liu (2008, lemma 2.2) and (H2) as well as (H3), we can show that for some C4>0,(3.4)Hence, substituting (3.2)–(3.4) into (3.1) implies thatIf T>0 is sufficiently small, then we can ensure thatConsequently, from the contraction mapping theorem, we derive that has a unique fixed point that is the mild solution to equation (2.1) on t∈[0,T]. Then repeating the procedure above, we can show equation (2.1) admits a unique mild solution. ▪

Since the mild solutions do not have stochastic differentials, by the Itô formula, we cannot deal with mild solutions directly in most arguments. For any t≥0, we introduce the following approximating system:(3.5)Here nρ(A), the resolvent set of A and R(n)=nR(n,A), R(n,A) is the resolvent of A. Similar to operator defined in theorem 2.4, the operator n associated with (3.5), for any x, y(A), can be defined as follows:

Theorem 3.2

Let be an arbitrarily given initial datum and assume that conditions (H1), (H2) and (H3) hold. Then (3.5) has a unique strong solution , which lies in for all T>0. Moreover, Xn(t) converges to the mild solution X(t) of equation (2.1) almost surely in as n→∞.

Proof

Since the proof is similar to that of Luo & Liu (2008, proposition 2.4), for the self-contained, we here give only the sketch of the proof. By theorem 3.1, for any t≥0, (3.5) admits a unique mild solution denoted by Xn(t). To show the first assertion, it is sufficient to show that the mild solution Xn(t) of (3.5) is also its unique strong solution. Since the operator AR(n)=AnR(n,A)=nn2R(n,A) is bounded, for any t≥0 it is easy to show almost surelyandThus, by Fubini theorem and property of semigroupNow using stochastic Fubini theorem (Ichikawa 1982, proposition 1.8), computeWhile from Luo & Liu (2008, lemma 2.1), we arrive atWe then derive from (3.5) that for any t≥0,which implies AXn(t) is integrable almost surely and . Recalling that Xn(t) is the mild solution to (3.5), we therefore deduce by the definition of strong solution that the first assertion is true. In what follows, we claim that the second assertion holds. Compute from (2.1) and (3.5) that

Observe that ‖R(n)‖≤2 for large n. By the Hölder inequality and (H2), for sufficiently large n, we obtainSimilarly, applying Liu (2004, theorem 1.2.6, p. 14) and using the condition (H2), for sufficiently large n, there exists a constant β1>0 such thatNow, taking into account Luo & Liu (2008, lemma 2.2) and (H2), for sufficiently large n and certain positive constants β2,β3,Next, observing ‖IR(n)‖→0 as n→∞, together with (H2), it is easy to seeCombining these bounds together, we deduce that there are numbers C(T) and ϵ(n) such thatwhere . By the Gronwall inequality,The desired assertion then follows from the standard diagonal sequence arguments. ▪

Remark 3.3

Obviously, the mild solution Xξ(t) of equation (2.1) is not a Markov process as it is dependent on the past. However, we can show that is a time-homogeneous strong Markov process as in Mohammed (1984). The Markov property will be used in remark 4.2 and proof of inequality (4.18).

## 4. Stability in distribution

In this section, we are concerned with the stability in distribution of mild solutions to equation (2.1). Now, we recall the definition.

Let p(t,ξ,dζ) denote the transition probability of the process y(t) with the initial state y(0)=ξ. Denote by P(t,ξ,Γ) the probability of event {y(t)∈Γ} given initial condition y(0)=ξ, i.e. with Γ(H), which denotes the Borel σ-algebra of H,Denote by Xξ(t) the mild solution to equation (2.1) with the initial datum . Correspondingly, .

Definition 4.1

The process is said to be stable in distribution if there exists a probability measure π(.) on D([−τ,0];H) such that its transition probability p(t,ξ,dζ) converges weakly to π(dζ) as t→∞ for every . In this case, equation (2.1) is said to be stable in distribution.

Remark 4.2

Since is a Markov process, using the Kolmogorov–Chapman equation, it is not difficult to show that the stability in distribution of implies the existence of a unique invariant probability measure for .

For our stability analysis, we need to prepare for several lemmas below.

Lemma 4.3

Let the conditions of theorem 3.2 hold. Assume there exist constants λ1>λ2≥0 and β≥0 such that, for any ,(4.1)Then(4.2)

Proof

For simplicity, we denote Xξ(t) by X(t). Firstly, we show that, for any t≥0,(4.3)For any t≥0, μ>0, using (4.1) and applying the Itô formula to the function and the strong solution Xn(t) of (3.5), we haveNoting that for λ1>λ2≥0 the equationadmits a unique positive root denoted by ρ. Letting μ=ρ and using theorem 3.2 and the dominated convergence theorem, we getThus, together with , the desired assertion (4.3) follows. Next, we intend to show that, for any t≥0,(4.4)Again, applying Itô's formula to the function and the strong solution Xn(t) of (3.5), for any tτ and θ∈[−τ,0], we obtainBy theorem 3.2, together with the dominated convergence theorem, we thus get(4.5)Now, by virtue of Burkholder–Davis–Gundy's inequality (Ichikawa 1982, proposition 1.6), we derive for some positive constant K1 such thatwhich, by using the Hölder's inequality, immediately yields for certain K2>0(4.6)Next, we shall estimate the last term of (4.5). The method used here is similar to that of the proof (Röckner & Zhang 2007, proposition 3.1). Letand [M,M]t,0 denote the quadratic variation of the process M(t,0). Next, from Burkholder–Davis–Gundy's inequality (Applebaum 2004, p. 235), there exists a positive constant K3 such that(4.7)By the definition of quadratic variation,Hence, in (4.7)(4.8)Substituting (4.3), (4.6), (4.8) into (4.5) and combining (H2), directly we have that (4.4) holds. Therefore, the desired assertion (4.2) follows. ▪

Remark 4.4

By the well-known Chebyshev inequality, for any positive number l, we haveLet l→∞, (4.2) implies that the right-hand side tends to 0. Therefore, for any ϵ>0, there is a compact subset of D([−τ,0];H) such that . That is, the family {p(t, ξ, dζ):t≥0} is tight.

In what follows, we consider the difference of two mild solutions that start from different initial conditions, namely(4.9)Furthermore, we introduce an approximating system in correspondence with (4.9) in the following form:(4.10)where nρ(A), the resolvent set of A and R(n)=nR(n,A), R(n,A) is the resolvent of A.

For given UC2(H;R+), define an operator nU:H4R associated with (4.10) by for any

Lemma 4.5

Suppose the conditions of theorem 3.2 hold. Assume also that there are constants λ3>λ4≥0 such that, for any ,(4.11)Then, for any compact subset of D([−τ,0];H),(4.12)

Proof

For integer NR+ and the strong solution to approximating system (4.10), we define stopping time as follows:Clearly, τN→∞ almost surely as N→∞. Let TNNt. Using the Itô formula to the strong solution of (4.10), for any t≥0 and λ>0, we derive(4.13)Noting that for λ3>λ4≥0, the following equationadmits a unique positive root denoted by δ. In (4.13), letting λ=δ and using theorem 3.2 and the dominated convergence theorem, it thus follows:That is,Hence, for any ϵ>0, there exists a δ>0 such that for ‖ξηD<δ(4.14)and(4.15)Since is compact, there exist ξ1,ξ2,…,ξk such that , where . By (4.15), there exists a T1>0 such that for tT1 and 1≤u, vk,(4.16)For any , we can find l, m such that . By (4.14) and (4.16), we derive thatConsequently, for any compact subset of D([−τ,0];H),In the sequel, in the same way as lemma 4.3 was proved, we can deduce the desired assertion. ▪

Remark 4.6

Under the conditions of lemma 4.5, we can show that for any ϵ>0 and any compact subset of D([−τ,0];H), there is a such that for any Actually, it is sufficient to show that there exists such that for any tTBy the Chebyshev inequality,However, by virtue of lemma 4.5, it follows that there exists such that for any tTthen the required assertion follows.

Let (D([−τ,0];H)) denote all probability measures on D([−τ,0];H). For P1,P2(D([−τ,0];H)) define metric dL as follows:and

Lemma 4.7

Let (4.2) and (4.12) hold. Then, {p(t,ξ,.):t≥0} is Cauchy in the space (D([−τ,0];H)) for any .

Proof

Fix . We need to show that, for any ϵ>0, there exists a T>0 such thatwhich is equivalent to show that for any fL(4.17)We thus compute for any fL and t, s>0(4.18)where for the first equality we used the property of conditional expectation, while the Markov property of has been used in the second equality. From remark 4.4, there exists a compact subset of D([−τ,0];H) for ϵ>0 such that(4.19)Using (4.18) and (4.19), we obtain(4.20)Furthermore, from (4.12), we derive that there is a T>0 for the given ϵ>0 such that(4.21)which, in addition to (4.20), impliesSince fL is arbitrary, the desired inequality (4.17) is obtained. ▪

Lemma 4.8

Let (4.12) hold. Then for any compact subset of D([−τ,0];H),

Proof

We need to show that, for any ϵ>0 and , there is a T>0 such thatwhich is equivalent to, for any ,As a matter of fact, for any fL,From (4.12), for any , there exists a T>0 satisfyingSince fL is arbitrary, we obtain thatThe desired result is now proved. ▪

After the preparation of the lemmas above, we now present our main result.

Theorem 4.9

Under the conditions of lemmas 4.3 and 4.5, the mild solution Xξ(t) to equation (2.1) is stable in distribution.

Proof

By the definition of stability in distribution, we need to show that there exists a probability measure π(.) such that for any , the transition probabilities {p(t, ξ,.):t≥0} converge weakly to π(.). As we know, the weak convergence of probability measures is equivalent to a metric concept, we then need to show that, for any ,

By lemma 4.7, {p(t, 0,.):t≥0} is Cauchy in the space (D([−τ,0];H)) with metric dL. Since (D([−τ, 0];H)) is a complete metric space under metric dL (cf. Chen 1992, theorem 5.4), there is a unique probability measure π(.)∈(D([−τ,0];H)) such that(4.22)

Moreover,

So, an application of lemma 4.8 yields that

## 5. An illustrative example

Consider the stochastic process Z(t,x) with Poisson jumps described bywhere B(t),t≥0, is a real standard Brownian motion and is a compensated Poisson random measure on [1,∞] with parameter λ(dy)dt such that . Assume moreover that B(t) is independent of . f is a real Lipschitz continuous function on L2(0,π) satisfying for u,vL2(0,π)with some positive constants c, k. In this example, we take H=L2(0,π) and with domain . It is easy to show that for arbitrary Moreover, for any ,Similarly, for ,Therefore, if and , by theorem 4.3, we then immediately observe that the mild solution process Z(t,x) is stable in distribution.

## Acknowledgments

The authors would like to thank the referees and the associated editor for their useful comments and suggestions.