Small noise asymptotic expansions for stochastic PDE's, I. the case of a dissipative polynomially bounded nonlinearity

Abstract

We study a reaction-diffusion evolution equation perturbed by a Gaussian noise. Here the leading operator is the infinitesimal generator of a C 0-semigroup of strictly negative type, the nonlinear term has at most polynomial growth and is such that the whole system is dissipative. The corresponding Itô stochastic equation describes a process on a Hilbert space with dissipative nonlinear, non globally Lipschitz drift and a Gaussian noise. Under smoothness assumptions on the nonlinearity, asymptotics to all orders in a small parameter in front of the noise are given, with uniform 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 example we consider the small noise asymptotic expansions for the stochastic FitzHugh-Nagumo equations of neurobiology around deterministic solutions.