Dirichlet problems and exit distributions for the telegraph process and its planar extensions

TL;DR

This paper analyzes boundary-value problems for the telegraph process and its planar extensions, deriving equations for exit distributions and mean exit times, and connecting results to Brownian motion in the hydrodynamic limit.

Contribution

It introduces new Dirichlet problems for finite-velocity models and extends analysis to planar cases with explicit solutions in certain geometries.

Findings

Derived Dirichlet problems for the telegraph process with and without drift.

Extended analysis to planar finite-velocity models with explicit solutions.

Showed convergence of equations to Brownian motion results in the hydrodynamic limit.

Abstract

In this paper, we study boundary-value problems describing the exit distribution of finite-velocity random motions from prescribed domains. For the standard telegraph process, with and without drift, we derive the Dirichlet problems governing the exit point and mean exit time from a closed interval. We then extend the analysis to a planar finite-velocity model with orthogonal directions, for which we obtain the associated Laplace and Poisson-type equations for the exit distribution and mean exit time. In the special case of an infinite strip, explicit solutions are obtained. In all cases, we show that our equations and results converge, in the hydrodynamic limit, to the corresponding ones for Brownian motion.