First Exit Times of Harmonically Trapped Particles: A Didactic Review

TL;DR

This paper provides a comprehensive review of the mathematical analysis of first exit times for harmonically trapped particles modeled by Ornstein-Uhlenbeck processes, including derivations, series representations, and applications.

Contribution

It offers a didactic overview with new series representations for confluent hypergeometric functions and discusses diverse applications and extensions of first exit time analysis.

Findings

Derived explicit formulas for mean exit time and survival probability.

Introduced a rapidly converging series for numerical computation.

Demonstrated applications in biophysics, finance, and physics.

Abstract

We revise the classical problem of characterizing first exit times of a harmonically trapped particle whose motion is described by one- or multi-dimensional Ornstein-Uhlenbeck process. We start by recalling the main derivation steps of a propagator using Langevin and Fokker-Planck equations. The mean exit time, the moment-generating function, and the survival probability are then expressed through confluent hypergeometric functions and thoroughly analyzed. We also present a rapidly converging series representation of confluent hypergeometric functions that is particularly well suited for numerical computation of eigenvalues and eigenfunctions of the governing Fokker-Planck operator. We discuss several applications of first exit times such as detection of time intervals during which motor proteins exert a constant force onto a tracer in optical tweezers single-particle tracking experiments; adhesion bond dissociation under mechanical stress; characterization of active periods of trend following and mean-reverting strategies in algorithmic trading on stock markets; relation to the distribution of first crossing times of a moving boundary by Brownian motion. Some extensions are described, including diffusion under quadratic double-well potential and anomalous diffusion.