Large Deviation Principle for Semilinear Stochastic Evolution Equations with Monotone Nonlinearity and Multiplicative Noise

TL;DR

This paper establishes a large deviation principle for solutions of semilinear stochastic evolution equations with monotone nonlinearities and multiplicative noise, using weak convergence methods and Itô inequalities.

Contribution

It introduces a novel application of weak convergence techniques to prove large deviation principles for a broad class of stochastic evolution equations with monotone nonlinearities.

Findings

Large deviation principle proven for semilinear stochastic evolution equations.

Application of weak convergence method in this context.

Illustrative examples demonstrating the theorems' applicability.

Abstract

We demonstrate the large deviation property for the mild solutions of stochastic evolution equations with monotone nonlinearity and multiplica- tive noise. This is achieved using the recently developed weak convergence method, in studying the large deviation principle. An It^o-type inequality is a main tool in the proofs. We also give two examples to illustrate the applications of the theorems.