Tohoku Mathematical Journal
2011
December
SECOND SERIES VOL. 63, NO. 4

Tohoku Math. J.
63 (2011), 877-898

Title SMALL NOISE ASYMPTOTIC EXPANSIONS FOR STOCHASTIC PDE'S, I. THE CASE OF A DISSIPATIVE POLYNOMIALLY BOUNDED NON LINEARITY

Author Sergio Albeverio, Luca Di Persio and Elisa Mastrogiacomo

(Received November 30, 2010, revised September 1, 2011)
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.

2000 Mathematics Subject Classification. Primary 35K57; Secondary 92B20, 35R60, 35C20.
Key words and phrases. Reaction-diffusion equations, dissipative systems, asymptotic expansions, polynomially bounded nonlinearity, stochastic FitzHugh-Nagumo system.

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