Estimating Small Probabilities for Langevin Dynamics

TL;DR

This paper develops a new method for estimating very small transition probabilities in Langevin dynamics by simplifying Girsanov's formula and deriving asymptotic expressions, aiding understanding of rare events in stochastic processes.

Contribution

It introduces a simplified Girsanov's formula and derives asymptotic transition probability densities for Langevin dynamics, advancing theoretical tools for rare event estimation.

Findings

Derived an explicit relationship between the generator and measure change.

Obtained asymptotic expressions for transition probabilities.

Discussed estimation of escape probabilities from potential wells.

Abstract

The problem of estimating small transition probabilities for overdamped Langevin dynamics is considered. A simplification of Girsanov's formula is obtained in which the relationship between the infinitesimal generator of the underlying diffusion and the change of probability measure corresponding to a change in the potential energy is made explicit. From this formula an asymptotic expression for transition probability densities is derived. Separately the problem of estimating the probability that a small noise Langevin process excapes a potential well is discussed.