Large deviations for stochastic PDE with Levy noise

TL;DR

This paper establishes a large deviation principle for solutions of stochastic evolution equations with Levy noise, using viscosity solutions and control methods, and applies results to hyperbolic equations with subordinated Wiener processes.

Contribution

It introduces a novel approach combining viscosity solutions and control theory to analyze large deviations in stochastic PDEs with Levy noise.

Findings

Proved a large deviation principle for Levy-driven stochastic PDEs.

Identified the Laplace limit as a viscosity solution of a Hamilton-Jacobi-Bellman equation.

Established exponential moment estimates for solutions with Levy noise.

Abstract

We prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small Levy noise. We use general large deviations theorems of Varadhan and Bryc, viscosity solutions of integro-partial differential equations in Hilbert spaces, and deterministic optimal control methods. The Laplace limit is identified as a viscosity solution of a Hamilton-Jacobi-Bellman equation of an associated control problem. We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Levy noise. General results are applied to stochastic hyperbolic equations perturbed by subordinated Wiener process.