Large deviation limits of invariant measures

TL;DR

This paper establishes conditions under which the large deviation principle for invariant measures of stochastic processes can be derived from the sample path LDP, with applications to jump diffusions.

Contribution

It demonstrates that sample path LDP and exponential tightness imply the LDP for invariant measures, expanding understanding of large deviations in stochastic processes.

Findings

LDP for invariant measures follows from sample path LDP under certain conditions

Application to jump diffusions provides concrete examples

Uses large deviation convergence and idempotent probability methods

Abstract

This paper is concerned with the general theme of relating the Large Deviation Principle (LDP) for the invariant measures of stochastic processes to the associated sample path LDP. It is shown that if the sample path deviation function possesses certain structure and the invariant measures are exponentially tight, then the LDP for the invariant measures is implied by the sample path LDP, no other properties of the stochastic processes in question being material. As an application, we obtain an LDP for the stationary distributions of jump diffusions. Methods of large deviation convergence and idempotent probability play an integral part.