The exponential resolvent of a markov process and large deviations for markov processes via hamilton-jacobi equations

We study the Hamilton-Jacobi equation f − λHf = h, where Hf = 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 equa-tion, 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).