A note on stability of SPDEs driven by α-stable noises

Jiying Wang,Yulei Rao

doi:10.1186/1687-1847-2014-98

Abstract

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. MSC:60H15, 60G15.

Highlights

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 [, ], Chow [ ], Liu [ ] and Yuan and Bao [ ], to name a few, where the noise processes are assumed to be square integrable
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 α/ -subordinated cylindrical Brownian motion with α ∈ (, ), 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

Summary

Introduction

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 [ , ], Chow [ ], Liu [ ] and Yuan and Bao [ ], to name a few, where the noise processes are assumed to be square integrable. There are few papers on the asymptotic behavior of mild solution of SPDEs driven by α-stable processes. We shall discuss the stability property of mild solutions of a class of SPDEs driven by α-stable processes to close the gap.

Results

Conclusion