On the Hill relation and the mean reaction time for metastable processes

TL;DR

This paper explores how the Hill relation and quasi-stationary distributions can be used to analyze biasing errors in numerical methods for estimating mean reaction times in metastable Markov processes, with applications to molecular dynamics.

Contribution

It introduces a theoretical framework for bias error analysis in computing mean reaction times, applicable to elliptic diffusions and various numerical procedures.

Findings

The Hill relation effectively characterizes reaction times.

Biasing errors can be sharply estimated using the proposed analysis.

Applications demonstrate the method's broad applicability.

Abstract

We illustrate how the Hill relation and the notion of quasi-stationary distribution can be used to analyse the biasing error introduced by many numerical procedures that have been proposed in the literature, in particular in molecular dynamics, to compute mean reaction times between metastable states for Markov processes. The theoretical findings are illustrated on various examples demonstrating the sharpness of the biasing error analysis as well as the applicability of our study to elliptic diffusions.