Stability in distribution of neutral stochastic partial differential delay equations driven by α-stable process
© Zang and Li; licensee Springer. 2014
Received: 15 September 2013
Accepted: 16 December 2013
Published: 9 January 2014
We consider a class of neutral stochastic partial differential equations driven by an α-stable process. We prove the existence and uniqueness of the mild solution to the equation by the Banach fixed-point theorem under some suitable assumptions. Sufficient conditions for the stability in the distribution of the mild solution are derived.
The theory of stochastic partial differential equations has been widely applied in scientific fields such as physics, mechanical engineering, and economics. Especially, the study of stochastic neutral functional differential equations has received a great deal of attention in recent years. For example, Bao et al.  extended the existence and uniqueness of mild solutions to a class of more general stochastic neutral partial functional differential equations under non-Lipschitz conditions. Caraballo et al.  investigated the exponential stability and ultimate boundedness of the solutions to a class of neutral stochastic semilinear partial delay differential equations.
Also the stability in a distribution is an important notion like the stability in probability or in the moment of stochastic differential equations. Such a stability is much weaker than stability in probability and it is useful sometimes to know whether or not the probability distribution of the solution will converge to some distribution but not necessarily to zero. There is an extensive literature concerned with the stability in the distribution of stochastic differential equations. Using an excellent stopping time technique and an M-matrix trick, Yuan and Mao  investigated the stability in the distribution of nonlinear SDEs with Markovian switching. Yuan et al.  discussed a class of stochastic differential delay equation with Markovian switching, where the sufficient conditions of stability in the distribution were established. Tan et al.  considered weak convergence of functional SDEs with variable delays. For the case of stochastic partial differential equations, we refer to Bao et al. [1, 6]. Furthermore, for the nonlinear regime switching jump diffusion, we can refer to Yang and Yin .
Although many scholars have discussed the stability in the distribution of SDEs or functional SDEs where the noises are Brownian motion and jumps, the methods applied therein will not work if the considered noises are α-stable processes. As we know, for an α-stable process (), it only has a moment. Therefore, some useful techniques involved in the above references, such as the Burkholder-Davis-Gundy inequality and Da Prato-Kwapien-Zabczyk’s factorization technique , are not available. On the other hand, it seems that little is known about the stability in the distribution of the neutral stochastic partial differential equations driven by an α-stable process, and there are few systematic works so far in which the noise source is an α-stable process as well. For more studies of stochastic systems driven by stable processes, we refer to [9–11].
where is the space of all càdlág functions paths from into H, a Hilbert space, equipped with the supremum norm . And are given functions to be specified later.
The contents of the paper are as follows. In Section 2, we briefly present some basic notations and preliminaries. In Section 3 the existence and uniqueness of mild solutions are proved. In the last section, we devote to give the sufficient conditions of the stability in the distribution of the mild solution to Eq. (1).
Let be a real separable Hilbert space. Denote by the space of all H-valued càdlág functions defined on equipped with the uniform norm . Recall that a path is called càdlág if it is right-continuous having finite left-hand limits.
has independent increments;
for any ,
(H1) The operator is a self-adjoint compact operator on the Hilbert space H which is separable such that −A has discrete spectrum with corresponding eigenbasis of H. In this case A generates a compact -semigroup , , such that .
(H2) There exists a positive constant such that for all
(H3) There exist and a positive constant such that for all and
The constants and k satisfy .
(H4) There exists such that and .
The following two lemmas will play an important role in proving our main results. So let us state them now.
Lemma 2.1 
Lemma 2.2 
where , is a cylindrical α-stable process having the form of (2) and C depends on α, θ, β, p.
Remark 2.1 In Lemma 2.2, the constant β satisfies so that the convolutions are in H. Moreover, there exist some such that .
3 Existence and uniqueness
- (b)For arbitrary(3)
We have the following result.
The required assertion follows if we show that the operator Λ has a fixed point in the space by the Banach fixed-point theorem. We divide the proof into two steps.
which is inequality (4). The solution can be extended to the entire interval in finite steps by repeating the above procedure on intervals with such that . This completes the proof. □
4 Asymptotic stability in the distribution
In this section, we shall derive some sufficient conditions on the stability in the distribution for the process on . We need to introduce some more notations. For , let be the solution of Eq. (1) with initial datum . Correspondingly, on . As usual, is called the segment process of . Denote by the transition probability of the process , then is a time homogeneous Markov process according to Mohammed .
Now, we introduce the concept of stability in the distribution and prepare some useful lemmas as follows.
Definition 4.1 
The process is said to be asymptotically stable in the distribution if there exists a probability measure on such that the transition probability function of converges weakly to as for every . Equation (1) is said to be asymptotically stable in the distribution if the solution process is asymptotically stable in the distribution.
(N2) , for every .
In what follows, we need to prepare some useful lemmas to demonstrate that (N1) and (N2) hold under some imposed conditions.
Proof The proof is similar to Lemma 3.1 of , so we omit it here. □
for any .
Then the desired assertion (13) follows immediately from (22). □
uniformly in .
This obviously contradicts (27). Therefore, (23) holds. □
Lemma 4.4 Let assumptions (H1)-(H4), (13), and (23) hold. Then, for any initial data , is Cauchy in the space with the metric .
Since f is arbitrary, the desired inequality (28) must hold. □
Based on the results above, we can now state our main result.
Theorem 4.1 Let the assumptions (H1)-(H4) and (12) hold; then the process is stable in the distribution.
as required. □
To demonstrate the applications of Theorem 4.1, we give an illustrative example, motivated by [, Example 4.1].
where , , . Let , where and is an independent, real-valued, normalized, symmetric α-stable process sequence. It is trivial to see that . Due to , we have . Hence there exists such that , and therefore . In other words, the assumption (H4) holds for such a case.
where and , which is also a complete orthonormal system of H. Hence, (H3) holds. Consequently, by Theorem 3.1, there exists a unique mild solution to Eq. (33). In addition, Let , , , , ; if , then by Theorem 4.1, Eq. (33) is stable in the distribution.
We are very grateful to the anonymous referees and the associate editor for their careful reading and helpful comments. This work was supported by the National Natural Sciences Foundation of China (No. 11071259, 11371374), Research Fund for the Doctoral Program of Higher Education of China (No. 20110162110060).
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