Small Noise Asymptotic Expansions for Stochastic Pde’s.i the Case of a Dissipative Polynomially Bounded Non Linearity
نویسنده
چکیده
We study a reaction-diffusion evolution equation perturbed by a Gaussian noise. Here the leading operator is the infinitesimal generator of a C0-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 drift and a Gaussian noise. Under smoothness assumptions on the non-linearity, 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.
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