Mean first passage times of higher-dimensional velocity jump processes

TL;DR

This paper develops a comprehensive framework for estimating mean first passage times of velocity jump processes in higher dimensions, accounting for directional bias and persistence effects, with analytical and numerical validation.

Contribution

It introduces a universal formula for MFPT in velocity jump processes with various angular distributions and explores anomalous scaling in narrow capture limits.

Findings

Derived a universal MFPT form for velocity jump processes with broad angular distributions.

Identified regimes where directional persistence leads to finite MFPT despite small targets.

Validated analytical predictions through numerical simulations.

Abstract

First passage phenomena arise across physics, biology, and finance when stochastic processes first reach a threshold, triggering downstream events. Examples include the irreversible exit from a domain, a biochemical reaction, a financial selloff. While typical formulations involve diffusive motion, many stochastic processes are better described as velocity jump processes, characterized by persistent motion interrupted by stochastic velocity changes. Despite their ubiquity, first-passage properties of velocity jump processes remain underdeveloped in higher dimensions, especially under directional bias. We present a general framework to estimate the mean first passage time (MFPT) and higher moments of the survival probability for fixed-speed velocity jump processes where possible reorientations range from strong alignment to full angular anisotropy. For low Knudsen numbers, when the mean free path is small compared to the distance to the target, we derive a universal form for the MFPT in which two bias functions encode broad classes of angular distributions, including von Mises-Fisher, wrapped Cauchy, and elliptical families. In the narrow capture limit of a vanishingly small target, directional persistence induces anomalous scaling, including regimes where the MFPT remains finite whereas standard diffusion would predict divergence. Finally, we obtain a Langevin representation that accurately reproduces first-passage statistics. Analytical predictions are confirmed by numerical simulations.