On properties of Continuous-Time Random Walks with Non-Poissonian jump-times

TL;DR

This paper generalizes the theory of continuous-time random walks (CTRWs) to arbitrary present times using renewal theory, extending beyond the traditional assumption that the present is a jump time, with detailed analysis of Erlang distributions.

Contribution

It introduces a generalized framework for CTRWs at arbitrary times, expanding the theoretical understanding beyond the Poissonian jump-time assumption.

Findings

Derived integral equations for the propagator at arbitrary times

Analyzed the case of Erlang distributed waiting times in detail

Provided several concrete examples illustrating the generalized theory

Abstract

The usual development of the continuous-time random walk (CTRW) proceeds by assuming that the present is one of the jumping times. Under this restrictive assumption integral equations for the propagator and mean escape times have been derived. We generalize these results to the case when the present is an arbitrary time by recourse to renewal theory. The case of Erlang distributed times is analyzed in detail. Several concrete examples are considered.