Large Deviations for a Class of Parabolic Semilinear Stochastic Partial Differential Equations in Any Space Dimension

TL;DR

This paper establishes a large deviation principle for solutions to a broad class of parabolic semilinear stochastic PDEs with multiplicative noise, applicable in any spatial dimension and for nonlinearities of arbitrary polynomial growth.

Contribution

It extends large deviation results to high-dimensional, nonlinear stochastic PDEs with multiplicative noise using the weak convergence approach.

Findings

Large deviation principle proven for solutions in $C([0,T]:L^ ho(D))$

Applicable to equations with polynomial nonlinearities of any order

Valid in any space dimension with smooth bounded domains

Abstract

We prove the large deviation principle for the law of the solutions to a class of parabolic semilinear stochastic partial differential equations driven by multiplicative noise, in $C\big([0,T]:L^\rho(D)\big)$, where $D\subset {\mathbb R}^d$ with $d\geqslant 1$ is a bounded convex domain with smooth boundary and $\rho$ is any real, positive and large enough number. The equation has nonlinearities of polynomial growth of any order, the space variable is of any dimension, and the proof is based on the weak convergence method.