Two mathematical tools to analyze metastable stochastic processes

TL;DR

This paper reviews mathematical tools like entropy estimates, logarithmic Sobolev inequalities, and quasi-stationary distributions to analyze metastability in stochastic processes, with applications to sampling algorithms and coarse-graining.

Contribution

It introduces the application of these mathematical tools to quantify metastability and evaluate algorithm efficiency in overdamped Langevin dynamics.

Findings

Entropy estimates and inequalities help quantify metastability.

Quasi-stationary distributions assist in analyzing metastable states.

Tools improve understanding of sampling algorithms and coarse-graining errors.

Abstract

We present how entropy estimates and logarithmic Sobolev inequalities on the one hand, and the notion of quasi-stationary distribution on the other hand, are useful tools to analyze metastable overdamped Langevin dynamics, in particular to quantify the degree of metastability. We discuss the interest of these approaches to estimate the efficiency of some classical algorithms used to speed up the sampling, and to evaluate the error introduced by some coarse-graining procedures. This paper is a summary of a plenary talk given by the author at the ENUMATH 2011 conference.