Mean first passage times of higher-dimensional velocity jump processes

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.

References in corpus (15)

• Probing microscopic origins of confined subdiffusion by first-passage observables
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• First-passage time of run-and-tumble particles
• Random walks of molecular motors arising from diffusional encounters with immobilized filaments
• The escape problem under stochastic volatility: the Heston model
• Mechanisms of receptor/coreceptor-mediated entry of enveloped viruses
• Velocity alignment leads to high persistence in confined cells
• The random walk of intermittently self-propelled particles
• Fluctuating landscapes and heavy tails in animal behavior
• Fast rare events in exit times distributions of jump processes
• A probabilistic model of relapse in drug addiction
• Leftward, Rightward and Complete Exit Time Distributions of Jump Processes
• Mean first passage time of active Brownian particles in two dimensions
• Mean First Passage Times for Transport Equations
• Langevin equations and a geometric integration scheme for the overdamped limit of rotational Brownian motion of axisymmetric particles