Moments of the first passage time under external driving

TL;DR

This paper develops a general theory for the moments of first passage times in one-dimensional Markov processes under weak external forcing, providing explicit formulas and validating them with simulations.

Contribution

It introduces a novel analytical framework for calculating first passage time moments under time-dependent forces, including explicit solutions for specific potentials and driving forces.

Findings

Analytical formulas agree well with simulations.

Exponential driving causes complex nonmonotonous effects.

Explicit expressions for mean and variance in key potentials.

Abstract

A general theory is derived for the moments of the first passage time of a one-dimensional Markov process in presence of a weak time-dependent forcing. The linear corrections to the moments can be expressed by quadratures of the potential and of the time-dependent probability density of the unperturbed system or equivalently by its Laplace transform. If none of the latter functions is known, the derived formulas may still be useful for specific cases including a slow driving or a driving with power at only small or large times. In the second part of the paper, explicite expressions for mean and variance of the first passage time are derived for the cases of a linear or a parabolic potential and an exponentially decaying driving force. The analytical results are found to be in excellent agreement with computer simulations of the respective first-passage processes. The particular examples furthermore demonstrate that already the effect of a simple exponential driving can be fairly involved implying a nontrivial nonmonotonous behavior of mean and variance as functions of the driving's time scale.