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Tohoku Mathematical Journal
SECOND SERIES VOL. 63, NO. 4
|Tohoku Math. J.|
63 (2011), 877-898
SMALL NOISE ASYMPTOTIC EXPANSIONS FOR STOCHASTIC PDE'S, I. THE CASE OF A DISSIPATIVE POLYNOMIALLY BOUNDED NON LINEARITY
Sergio Albeverio, Luca Di Persio and Elisa Mastrogiacomo
(Received November 30, 2010, revised September 1, 2011)
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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