Large deviation properties of the empirical measure of a metastable small noise diffusion

TL;DR

This paper develops large deviation approximations for the empirical measure of small noise diffusions, utilizing Freidlin-Wentzell theory and quasipotentials to analyze metastable states and transition rates, aiding Monte Carlo scheme design.

Contribution

It introduces new large deviation results for empirical measures of small noise diffusions, with applications to Monte Carlo algorithms and metastability analysis.

Findings

Large deviation limits for integrals over empirical measures are established.

First and second moments of integrals are expressed in terms of quasipotentials.

New techniques connect regenerative cycle estimates to empirical measure deviations.

Abstract

The aim of this paper is to develop tractable large deviation approximations for the empirical measure of a small noise diffusion. The starting point is the Freidlin-Wentzell theory, which shows how to approximate via a large deviation principle the invariant distribution of such a diffusion. The rate function of the invariant measure is formulated in terms of quasipotentials, quantities that measure the difficulty of a transition from the neighborhood of one metastable set to another. The theory provides an intuitive and useful approximation for the invariant measure, and along the way many useful related results (e.g., transition rates between metastable states) are also developed. With the specific goal of design of Monte Carlo schemes in mind, we prove large deviation limits for integrals with respect to the empirical measure, where the process is considered over a time interval whose length grows as the noise decreases to zero. In particular, we show how the first and second moments of these integrals can be expressed in terms of quasipotentials. When the dynamics of the process depend on parameters, these approximations can be used for algorithm design, and applications of this sort will appear elsewhere. The use of a small noise limit is well motivated, since in this limit good sampling of the state space becomes most challenging. The proof exploits a regenerative structure, and a number of new techniques are needed to turn large deviation estimates over a regenerative cycle into estimates for the empirical measure and its moments.