Applied Mathematics Department Colloquium - Maria D'Orsogna

Maria-Rita D'Orsogna; Department of Mathematics; California State University Northridge

First Passage Properties of Velocity Jump Processes

First passage problems study the time it takes for a stochastic process, such as a random walk, to reach a target for the first time. These problems arise in many applications across physics, biology, and finance, in which reaching a target can trigger irreversible downstream events such as domain exit, biochemical reactions, or financial selloffs. Classical formulations typically assume diffusive, continuous dynamics, leading to analytical expressions for the survival probability and the mean first passage time (MFPT) to the target. Many real-world stochastic phenomena, however, are more accurately described by velocity jump processes (VJPs), characterized by persistent, directed motion interrupted by random velocity changes. Despite their ubiquity, the first passage properties of VJPs remain understudied. In this talk, we will present a general framework for estimating first passage properties of VJPs with fixed speed and random reorientations that follow a given angular distribution, such as the von Mises-Fisher, wrapped Cauchy, or elliptical distribution. Asymptotic expressions are derived for the MFPT to a target in the low Knudsen number regime, where the mean free path is small compared to the distance to the target. Explicit solutions are obtained for VJPs in two- and three-dimensional circular domains under radial symmetry. Remarkably, the MFPT scaling in the narrow capture problem can differ substantially from the classical diffusive prediction.