Large Deviations for Stochastic Partial Differential Equations Driven by a Poisson Random Measure

TL;DR

This paper develops a large deviation framework for stochastic PDEs driven by Poisson random measures, addressing the challenges of infinite-dimensional systems with irregular solutions, and illustrates it with a pollutant spread model.

Contribution

It introduces a novel large deviation approach for infinite-dimensional stochastic PDEs with Poisson noise, overcoming regularity and approximation challenges.

Findings

Established a large deviation principle for a class of stochastic PDEs with Poisson noise.

Applied the theory to model pollutant spread in waterways.

Provided a variational representation approach for nonnegative functionals of PRM.

Abstract

Stochastic partial differential equations driven by Poisson random measures (PRM) have been proposed as models for many different physical systems, where they are viewed as a refinement of a corresponding noiseless partial differential equations (PDE). A systematic framework for the study of probabilities of deviations of the stochastic PDE from the deterministic PDE is through the theory of large deviations. The goal of this work is to develop the large deviation theory for small Poisson noise perturbations of a general class of deterministic infinite dimensional models. Although the analogous questions for finite dimensional systems have been well studied, there are currently no general results in the infinite dimensional setting. This is in part due to the fact that in this setting solutions may have little spatial regularity, and thus classical approximation methods for large deviation analysis become intractable. The approach taken here, which is based on a variational representation for nonnegative functionals of general PRM, reduces the proof of the large deviation principle to establishing basic qualitative properties for controlled analogues of the underlying stochastic system. As an illustration of the general theory, we consider a particular system that models the spread of a pollutant in a waterway.