A note on stability of SPDEs driven by α-stable noises
© Wang and Rao; licensee Springer. 2014
Received: 24 December 2013
Accepted: 18 March 2014
Published: 28 March 2014
In this paper, by the Minkovski inequality and the semigroup method we discuss the stability of mild solutions for a class of SPDEs driven by α-stable noise, and the methods are also generalized to deal with the stability of SPDEs driven by subordinated cylindrical Brownian motion and fractional Brownian motion, respectively.
The stability of stochastic partial differential equations (SPDEs) driven by Brownian motions or Lévy processes has been well established; see, e.g., Bao and Yuan , Bao et al. [2, 3], Chow , Liu  and Yuan and Bao , to name a few, where the noise processes are assumed to be square integrable. However, such restriction clearly rules out the interesting α-stable processes since Wiener noise and Poisson-jump noise have arbitrary finite moments, while α-stable noise only has finite p th moment for with . Recently, stochastic equations driven by α-stable processes, which have plenty of applications in physics due to the fact that the α-stable noise exhibits the heavy tailed phenomenon, e.g., Solomon et al. , receives great attention. For example, Priola and Zabczyk  gave a proper starting point on the investigation of structural properties of SPDEs driven by an additive cylindrical stable noise, Dong et al.  studied the ergodicity of stochastic Burgers equations driven by -subordinated cylindrical Brownian motion with , and Zhang  established the Bismut-Elworthy-Li derivative formula for stochastic differential equations (SDEs) driven by α-stable noise. For finite-dimensional cases, Wang  derived a gradient estimate for linear SDEs driven by α-stable noise, and Wang  established the functional inequalities for Ornstein-Uhlenbeck processes driven by α-stable noise by the sharp estimates of density function for rotationally invariant symmetric α-stable Lévy processes. However, there are few papers on the asymptotic behavior of mild solution of SPDEs driven by α-stable processes. In this note, we shall discuss the stability property of mild solutions of a class of SPDEs driven by α-stable processes to close the gap. Due to the fact that α-stable noise only has finite p th moment for and that the stochastic evolution does not admit a stochastic differential, which leads to the Itô formula being unavailable, then some new tricks need to be put forward to overcome the difficulties brought about by α-stable noise.
This note is organized as follows: in Section 2 we apply the Minkovski inequality and the semigroup method to investigate the exponential stability of mild solutions for a class of SPDEs driven by α-stable processes, and these tricks are extended to cope with the stability of mild solutions of SPDEs driven by subordinated cylindrical Brownian motion and fractional Brownian motion, respectively, in Section 3.
2 Stability of SPDEs driven by α-stable noise
Here is an orthonormal basis of ℍ, are independent, ℝ-valued, normalized, symmetric α-stable Lévy processes defined on the stochastic basis , and is a sequence of positive numbers.
with initial value .
(A1) is a self-adjoint compact operator on ℍ such that −A has discrete spectrum , with corresponding eigenbasis of ℍ. In this case, by [, Theorem 6.26, p.185] and [, Theorem 6.29, p.187], A generates a -semigroup , , such that , where denotes the usual operator norm.
(A2) and there exist and such thatand
where is a local integrable continuous function.
(A3) is a continuous function such that
is locally bounded, i.e., is bounded on the time interval for any , where is the sequence appearing in (2.1) and is the discrete spectrum of A.
Next we recall the following Minkowski inequality, which plays a key role in revealing the stability property of SPDEs driven by α-stable processes.
We now can state our main results in this section.
That is, the solution is exponentially stable in the pth moment with the Lyapunov exponent .
holds and the desired assertion (2.6) follows due to . □
In what follows, we establish an example to demonstrate that the condition (2.5) holds in many practical situations.
due to the increasing property of the spectrum . If holds, then both (A3) and (2.5) are satisfied. There are plenty of examples such that the condition holds, e.g., [, Example 4.4]. Moreover, if , , , , , where 1 is the identical operator on H and is Lipschitzian such that , then (A2) and (2.4) holds.
Remark 2.1 To reveal the stability property of the mild solution of (2.1), we replace (A3) by a little bit strong condition (2.5), although (2.1) admits a unique mild solution in finite-time horizon under (A1)-(A3).
where is defined as in (2.7). Then under appropriate conditions, we can also discuss the stability property of the mild solution of (2.2).
Thus, carrying out a similar argument to that of Theorem 2.2, we conclude by the previous method that the Lyapunov exponent is dependent on , and . However, by the technique introduced in the argument of Theorem 2.2, we find that the Lyapunov exponent is , which only is dependent on and .
3 Extension to SPDEs driven by subordinated cylindrical Brownian motions and fractional Brownian motions
where is an -stable subordinator independent of , is a sequence of real numbers, and is an orthonormal basis of ℍ. Note that , which satisfies for each , is a Lévy process by [, Theorem 1.3.25, p.56] and an α-stable process due to [, Proposition 1.3.27, p.58].
One of our main results in this section is as follows.
In other words, the solution is exponentially stable in the pth moment, where the Lyapunov exponent is −γ.
Then, by , we take such that and the desired assertion follows immediately. □
Remark 3.1 We remark that Example 2.3 still satisfies the condition (3.3).
and is an orthonormal basis of H. are standard Brownian motions for , admit self-similarity, possess Hölder continuity, but are not semi-martingales for ; see, e.g., [, Chapter 5]. To the best of our knowledge, there are essentially two different approaches to construct stochastic integrals with respect to the fractional Brownian motion, i.e., a path-wise approach and the Malliavin calculus; see, e.g., the monograph .
For more details of the stochastic integration with respect to the infinite-dimensional fractional Brownian motion, we refer to, e.g., Tindel et al. .
Note that the solution of (3.7) is even not a semi-martingale so that the Itô formula is unavailable.
The other main results in this section is as follows.
In other words, the solution is exponentially stable in mean square, where the Lyapunov exponent is −γ.
Taking (2.4), (3.9), and the uniform boundedness of into account and thus applying the Gronwall inequality yields the desired assertion. □
Remark 3.2 By the approach introduced in the previous section, we discuss the mean square exponential stability of (3.7) under the condition . However, Caraballo et al.  investigated the same problem under . In other words, we have improved some existing results in certain sense.
Remark 3.3 All the results in this paper can be further extended to functional SPDEs driven by α-stable noise and fractional Brownian motion, including variable delay and distributed delay, while we here omit such discussions since there are no technical problems. Furthermore, there are also some interesting problems for SPDEs driven by α-stable noise to be investigated, e.g., since (2.2) is non-autonomous, the mild solution is not a homogeneous Markov process, which makes the investigation of the stability in distribution of analytic solution, a weaker stability notion than exponential stability, and the corresponding numerical stability, very interesting. Such a topic will be reported in our forthcoming paper.
This research is supported by the Natural Science Foundation of China under Grant no. 71372063.
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