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Stability in distribution of neutral stochastic partial differential delay equations driven by α-stable process
Advances in Difference Equations volume 2014, Article number: 13 (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].
In this paper, we study the existence, uniqueness, and stability in the distribution of mild solutions for the following neutral stochastic differential equation with finite delay:
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.
Let be a cylindrical α-stable process, , defined by
Here is an orthonormal basis of H, are independent, real-valued, normalized, symmetric α-stable Lévy processes defined on stochastic basis , and is a sequence of positive numbers. Recall that a stochastic process is called an α-stable Lévy process if
has independent increments;
for any ,
where η stands for an α-stable random variable, which is uniquely determined by its characteristic function involving four parameters: , the index of stability; , the skewness parameter; , the scale parameter; , the shift, and which has the form
where for and for . We call η is strictly α-stable whenever , and if, in addition, , η is said to be symmetric α-stable. For a real-valued normalized (standard) symmetric α-stable Lévy process , , it has the characteristic function
Throughout the paper we impose the following assumptions:
(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 
Under (H1), for any and
and there exists such that for any
Lemma 2.2 
Let be arbitrary. For all and all , we have
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
Definition 3.1 An -adapted càdlág stochastic process , is called a mild solution of Eq. (1) if it has the following properties:
We shall denote by the Banach space of all of càdlág H-valued processes with initial data for , and
We have the following result.
Theorem 3.1 Suppose the assumptions (H1)-(H4) hold and let , . Then, for any initial datum , there exists a unique mild solution of Eq. (1) in and there exists a constant , independent of ξ, such that
Proof For arbitrary and , define an operator Λ on by that , , and
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.
Step 1. We show that for . It is trivial for the case . For , and for any fixed , using the trivial inequality: , here when , when , we have
From (H1), (H3), and the boundedness of for , we get
where is a constant only depending on k and p. For the second term, , using the assumption (H3) again, we have
We consider the third term, : apply Lemma 2.1, use the assumption (H3) and Hölder’s inequality, and we derive
Similarly, for the fourth term, , using Hölder’s inequality, we obtain
Noting that , and by Lemma 2.2, we immediately get
Thus, we derive from (5) to (10) that, for some constants , , ,
Step 2. We shall show that the mapping Λ is contractive. Let . For any fixed , we have
By the assumptions (H2), (H3), and Hölder’s inequality, we obtain
where on , and is a bounded constant. Hence, by the condition , choosing sufficiently small such that , we can conclude that Λ is contractive. Therefore, by the contraction principle, we have a mild solution of Eq. (1) on . Moreover with such a we get from (11) for the solution of Eq. (1)
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.
Lemma 4.1 Let be Borel measurable. If is a solution of the delay integral inequality, then
where , , , , and J are nonnegative constants. If for some constant and
then there are constants and such that
where γ and N satisfy
Proof The proof is similar to Lemma 3.1 of , so we omit it here. □
Lemma 4.2 Assume that (H1)-(H4) hold, and that the inequality
holds for , , where is the gamma function. Then
for any .
Proof First, we shall show . For , we obtain from (3) that
where () stands for the i th term behind the first inequality. It follows from Lemma 2.1 and the assumption (H3) that
where is an appropriate constant only depend on k and p. For the second term ,
Similarly, by using Lemma 2.1, (H3), and Hölder’s inequality, we obtain
By (H2) and Hölder’s inequality, we obtain
For the estimation of , by the argument of [, Theorem 4.4] and applying (H4), we have
where depends only on p. Thus, substituting (15)-(19) into (14), we get
where , , and
In the light of (12), holds. Furthermore, there exist , , , such that
Consequently, combining (12), (20) with Lemma 4.1, we arrive at
Next, we use the following trivial inequality: for any and ,
For any integer , similar to the above computations, one has
where is also a cylindrical stable process, C is a constant not depending on n and its value is not important and may change from one line to another. Hence, according to (21) and Lemma 2.2,
holds. In view of (iii) of (H3) and (21), there exist constants ϑ and M such that, for sufficient small ϵ, , and for any integer ,
Observe that for any , there exists an such that and
Then the desired assertion (13) follows immediately from (22). □
Lemma 4.3 Let the conditions of (H1)-(H4) and (12) hold. Then for any bounded subset K of ,
uniformly in .
Proof Since the argument is similar to Lemma 4.2, we only sketch the main proof to point out the difference with that of Lemma 4.2 here. First, we shall prove that
uniformly in . Following a similar argument to derive (20), we can get
where , and
As a result, noting the condition (12) and using [, Lemma 3.1], we derive from (24) that
where μ is a positive root of the equation and . Hence,
Now, for and , according to the fundamental inequality and assumptions (H1)-(H4), we arrive at
This, together with (25), yields
On the other hand, when ,
This obviously contradicts (27). Therefore, (23) holds. □
In what follows we aim to prove the stability in the distribution of Eq. (1). Denote by the space of all probability measures on . For , define
where, for any ,
Lemma 4.4 Let assumptions (H1)-(H4), (13), and (23) hold. Then, for any initial data , is Cauchy in the space with the metric .
Proof For any fixed , we need to show that for any , there is a such that
for any and . Now, for any and , we compute
where and . By the argument (13), there exists a positive number R sufficiently large for which
On the other hand, by Lemma 4.3, there exists a such that
Substituting (30), (31) into (29) yields
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.
Proof By Definition 4.1, it suffices to prove that there is a probability measure such that for any , the transition probabilities converge weakly to . According to the well-known fact that the weak convergence of probability measure is a metric concept (see ), we therefore need to show that, for any ,
By Lemma 4.4, is Cauchy in the metric space for any fixed . Therefore, there exists a probability measure such that
Furthermore, for any , Lemma 4.3 shows that
as required. □
To demonstrate the applications of Theorem 4.1, we give an illustrative example, motivated by [, Example 4.1].
Example 4.1 Let be Lipschitzian, i.e., there exists such that , . Assume further that is measurable such that and
Consider the following stochastic neutral partial functional differential equation:
with the Dirichlet boundary condition
and the initial condition
Let ; A is given by
where , , represent Sobolev spaces, and is the subspace of of all functions vanishing at 0 and π. Then we get
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.
For and , let
Then Eq. (33) can be rewritten in the form (1). Observe that A generates a strongly continuous semigroup , which is compact, analytic and self-adjoint, and
Thus (H1) holds and . Furthermore note that
which in particular yields . As a result, using , together with (32), (34), and Hölder’s inequality, we get
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.
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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).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
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Zang, Y., Li, J. Stability in distribution of neutral stochastic partial differential delay equations driven by α-stable process. Adv Differ Equ 2014, 13 (2014). https://doi.org/10.1186/1687-1847-2014-13
- neutral stochastic partial differential equation
- α-stable process
- mild solution
- stability in distribution