Exact calculation of the mean first-passage time of continuous-time random walks by nonhomogeneous Wiener-Hopf integral equations

TL;DR

This paper derives an exact integral equation for calculating the mean first-passage time of asymmetric continuous-time random walks, accounting for non-Markovian effects and jump-size distributions, with implications for boundary-related stochastic processes.

Contribution

It introduces a nonhomogeneous Wiener-Hopf integral equation for exact MFPT calculation, extending analysis beyond asymptotic limits and generalizing to non-Markovian walks.

Findings

MFPT depends on the entire jump-size distribution in the boundary-adjacent region.

MFPT is independent of jump-size distribution in the opposite direction to the boundary.

A length-scale determines when the MFPT transitions from universal to distribution-dependent behavior.

Abstract

We study the mean first-passage time (MFPT) for asymmetric continuous-time random walks in continuous-space characterised by waiting-times with finite mean and by jump-sizes with both finite mean and finite variance. In the asymptotic limit, this well-controlled process is governed by an advection-diffusion equation and the MFPT results to be finite when the advecting velocity is in the direction of the boundary. We derive a nonhomogeneous Wiener-Hopf integral equation that allows for the exact calculation of the MFPT by avoiding asymptotic limits and it emerges to depend on the whole distribution of the jump-sizes and on the mean-value only of the waiting-times, thus it holds for general non-Markovian random walks. Through the case study of a quite general family of asymmetric distributions of the jump-sizes that is exponential towards the boundary and arbitrary in the opposite direction, we show that the MFPT is indeed independent of the jump-sizes distribution in the opposite direction to the boundary. Moreover, we show also that there exists a length-scale, which depends only on the features of the distribution of jumps in the direction of the boundary, such that for starting points near the boundary the MFPT depends on the specific whole distribution of jump-sizes, in opposition to the universality emerging for starting points far-away from the boundary.