Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

TL;DR

This paper establishes a large deviation principle for stochastic evolution equations with monotone drift and small multiplicative noise, applying the results to various SPDEs using a weak convergence approach.

Contribution

It extends large deviation principles to a broad class of SPDEs with monotone drift and multiplicative noise, using a novel weak convergence proof technique.

Findings

Large deviation principle proven for stochastic evolution equations.

Application to reaction-diffusion, porous media, and p-Laplace equations.

Weak convergence approach effectively establishes the Laplace principle.

Abstract

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.