Small noise asymptotic expansions for stochastic PDE's driven by dissipative nonlinearity and Levy noise

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:123

Article

Albeverio, Sergio^1,3  Mastrogiacomo, Elisa²  Smii, Boubaker³ 

[1] Univ Bonn, Dept Appl Math, HCM, Bonn, Germany

[2] Univ Milano Bicocca, Dipartimento Stat & Metodi Quantitativi, I-20126 Milan, Italy

[3] King Fahd Univ Petr & Minerals, Dept Math & Stat, Dhahran 31261, Saudi Arabia

关键词: SPDEs;    Dissipative systems;    Levy processes;    Stochastic convolution with Levy processes;    Asymptotic expansions;    Polynomially bounded nonlinearity;    Stochastic FitzHugh-Nagumo system;    Small noise;    Levy space time noise;   

DOI  :  10.1016/j.spa.2013.01.013

来源: Elsevier

【 摘 要 】

We study a reaction-diffusion evolution equation perturbed by a space time Levy noise. The associated Kolmogorov operator is the sum of the infinitesimal generator of a C-0-semigroup of strictly negative type acting on a Hilbert space and a nonlinear term which has at most polynomial growth, is non necessarily Lipschitz and is such that the whole system is dissipative. The corresponding Ito stochastic equation describes a process on a Hilbert space with dissipative nonlinear, non globally Lipschitz drift and a Levy noise. Under smoothness assumptions on the nonlinearity, asymptotics to all orders in a small parameter in front of the noise are given, with detailed estimates on the remainders. Applications to nonlinear SPDEs with a linear term in the drift given by a Laplacian in a bounded domain are included. As a particular case we provide the small noise asymptotic expansions for the SPDE equations of FitzHugh-Nagumo type in neurobiology with external impulsive noise. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free