The one-dimensional asymmetric persistent random walk

TL;DR

This paper investigates a one-dimensional asymmetric persistent random walk, deriving exact solutions for its Green's function, scattering statistics, and first-passage time distribution, revealing phenomena like transition behaviors in escape probability and residence time.

Contribution

It introduces an exact analytical framework for anisotropic mean free paths in 1D persistent random walks, extending the classical telegrapher's equation to include asymmetry.

Findings

Exact Green's function derived for the asymmetric case

Distribution of first-passage time at the origin obtained

Transition phenomena observed in escape probability and residence time

Abstract

Persistent random walks are intermediate transport processes between a uniform rectilinear motion and a Brownian motion. They are formed by successive steps of random finite lengths and directions travelled at a fixed speed. The isotropic and symmetric one-dimensional persistent random walk is governed by the telegrapher's equation, also called hyperbolic heat conduction equation. These equations have been designed to resolve the paradox of the infinite speed in the heat and diffusion equations. The finiteness of both the speed and the correlation length leads to several classes of random walks: Persistent random walk in one dimension can display anomalies that cannot arise for Brownian motion such as anisotropy and asymmetries. In this work we focus on the case where the mean free path is anisotropic, the only anomaly leading to a physics that is different from the telegrapher's case. We derive exact expression of its Green's function, for its scattering statistics and distribution of first-passage time at the origin. The phenomenology of the latter shows a transition for quantities like the escape probability and the residence time. (Second version that corrects the typos of the first version).