Tohoku Mathematical Journal

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

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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.

Article information

Tohoku Math. J. (2), Volume 63, Number 4 (2011), 877-898.

First available in Project Euclid: 6 January 2012

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Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations
Secondary: 92B20: Neural networks, artificial life and related topics [See also 68T05, 82C32, 94Cxx] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35C20: Asymptotic expansions

Reaction-diffusion equations dissipative systems asymptotic expansions polynomially bounded nonlinearity stochastic FitzHugh-Nagumo system


Albeverio, Sergio; Di Persio, Luca; Mastrogiacomo, Elisa. Small noise asymptotic expansions for stochastic PDE's, I. The case of a dissipative polynomially bounded non linearity. Tohoku Math. J. (2) 63 (2011), no. 4, 877--898. doi:10.2748/tmj/1325886292.

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