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2011 Small noise asymptotic expansions for stochastic PDE's, I. The case of a dissipative polynomially bounded non linearity
Sergio Albeverio, Luca Di Persio, Elisa Mastrogiacomo
Tohoku Math. J. (2) 63(4): 877-898 (2011). DOI: 10.2748/tmj/1325886292

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.

Citation

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Sergio Albeverio. Luca Di Persio. Elisa Mastrogiacomo. "Small noise asymptotic expansions for stochastic PDE's, I. The case of a dissipative polynomially bounded non linearity." Tohoku Math. J. (2) 63 (4) 877 - 898, 2011. https://doi.org/10.2748/tmj/1325886292

Information

Published: 2011
First available in Project Euclid: 6 January 2012

zbMATH: 1234.35328
MathSciNet: MR2872967
Digital Object Identifier: 10.2748/tmj/1325886292

Subjects:
Primary: 35K57
Secondary: 35C20, 35R60, 92B20

Rights: Copyright © 2011 Tohoku University

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