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

摘要

We study a reaction-diffusion evolution equation perturbed by a space-time Lévy noise. The associated Kolmogorov operator is the sum of the infinitesimal generator of a ~(C0)-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 It? stochastic equation describes a process on a Hilbert space with dissipative nonlinear, non globally Lipschitz drift and a Lévy 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.