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.

Sergio, A., Luca, D., Mastrogiacomo, E. (2011). Small noise asymptotic expansions for stochastic PDE's. The case of a dissipative polynomiallly bounded non linearity. TOHOKU MATHEMATICAL JOURNAL, 63(4), 877-898 [10.2748/tmj/1325886292].

Small noise asymptotic expansions for stochastic PDE's. The case of a dissipative polynomiallly bounded non linearity

MASTROGIACOMO, ELISA
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.
Articolo in rivista - Articolo scientifico
Reaction-diffusion equations; dissipative systems; asymptotic expansions; polynomially bounded nonlinearity; stochastic FitzHugh-Nagumo system
English
2011
63
4
877
898
none
Sergio, A., Luca, D., Mastrogiacomo, E. (2011). Small noise asymptotic expansions for stochastic PDE's. The case of a dissipative polynomiallly bounded non linearity. TOHOKU MATHEMATICAL JOURNAL, 63(4), 877-898 [10.2748/tmj/1325886292].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/48439
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