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

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 an 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 (2006).