Large Deviations for Markov jump processes in periodic and locally periodic environments

TL;DR

This paper establishes large deviation principles for Markov jump processes in environments with periodic or locally periodic microstructures, linking the rate function to spectral problems, under ellipticity and localization conditions.

Contribution

It introduces a framework for analyzing large deviations of jump processes in complex microstructured environments using spectral problem formulations.

Findings

Large deviation principle proven for jump processes in periodic environments

Rate function characterized via auxiliary spectral problems

Applicable under ellipticity and localization assumptions

Abstract

The paper deals with a family of jump Markov process defined in a medium with a periodic or locally periodic microstructure. We assume that the generator of the process is a zero order convolution type operator with rapidly oscillating locally periodic coefficient and, under natural ellipticity and localization conditions, show that the family satisfies the large deviation principle in the path space equipped with Skorokhod topology. The corresponding rate function is defined in terms of a family of auxiliary periodic spectral problems.