Exit times in non-Markovian drifting continuous-time random walk processes

TL;DR

This paper derives equations for the mean exit time in non-Markovian drifting continuous-time random walks, highlighting the effects of non-Markovian properties and providing closed-form solutions for specific cases.

Contribution

It introduces a renewal theory-based framework for analyzing exit times in non-Markovian CTRWs with drift, including explicit solutions for certain distributions.

Findings

Closed-form solutions for exit times when drift and jumps share the same sign

Analysis of non-Markovian corrections to classical models

Detailed case study with Erlang-distributed holding times

Abstract

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.