The exponential resolvent of a Markov process and large deviations for Markov processes via Hamilton-Jacobi equations

TL;DR

This paper introduces a new operator as the exponential resolvent of a Markov process's Hamilton-Jacobi equation, linking it to large deviations via path-space relative entropy and providing a novel proof of existing large deviation results.

Contribution

It identifies a viscosity solution operator for the Hamilton-Jacobi equation associated with Markov processes, connecting it to large deviations through an optimization problem involving relative entropy.

Findings

Defined a new exponential resolvent operator for Markov processes.

Provided a new proof of Feng and Kurtz's large deviation theorem.

Linked Hamilton-Jacobi equations to large deviations via path-space entropy.

Abstract

We study the Hamilton-Jacobi equation f - lambda Hf = h, where H f = e^{-f}Ae^f and where A is an operator that corresponds to a well-posed martingale problem. We identify an operator that gives viscosity solutions to the Hamilton-Jacobi equation, and which can therefore be interpreted as the resolvent of H. The operator is given in terms of optimization problem where the running cost is a path-space relative entropy. Finally, we use the resolvents to give a new proof of the abstract large deviation result of Feng and Kurtz.