Small noise asymptotic expansions for stochastic PDE's driven by dissipative nonlinearity and L\'evy noise

TL;DR

This paper develops small noise asymptotic expansions for stochastic reaction-diffusion equations driven by Le9vy noise, providing detailed estimates and applications to nonlinear SPDEs, including models relevant to neurobiology.

Contribution

It introduces a method for deriving asymptotic expansions to all orders for SPDEs with dissipative nonlinearities and Le9vy noise, including detailed remainder estimates.

Findings

Asymptotic expansions are obtained for SPDEs with small Le9vy noise.

Applications include nonlinear SPDEs with Laplacian drift and neurobiological models.

Detailed estimates on remainders enhance understanding of noise effects.

Abstract

We study a reaction-diffusion evolution equation perturbed by a space-time L\'evy noise. The associated Kolmogorov operator is the sum of the infinitesimal generator of a $C_0$-semigroup of strictly negative type acting in a Hilbert space and a nonlinear term which has at most polynomial growth, is non necessarily Lipschitz and is such that the whole system is dissipative. The corresponding It\^o stochastic equation describes a process on a Hilbert space with dissi- pative nonlinear, non globally Lipschitz drift and a L\'evy noise. Under smoothness assumptions on the non-linearity, asymptotics to all orders in a small parameter in front of the noise are given, with detailed 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 case we provide the small noise asymptotic expansions for the SPDE equations of FitzHugh Nagumo type in neurobiology with external impulsive noise.