Closed-form solutions for continuous time random walks on finite chains

TL;DR

This paper derives closed-form Laplace space solutions for Green's functions and first passage time PDFs of continuous time random walks on finite chains, using a novel trajectory counting technique, applicable to inhomogeneous systems.

Contribution

Introduces a new trajectory counting method to obtain explicit solutions for CTRWs on finite chains, linking Green's functions and higher order propagators.

Findings

Closed-form Laplace solutions for Green's functions.

Explicit formulas for first passage time PDFs.

Application to escape problems from biased chains.

Abstract

Continuous time random walks (CTRW) on finite arbitrarily inhomogeneous chains are studied. By introducing a technique of counting all possible trajectories, we derive closed-form solutions in Laplace space for the Green's function and for the first passage time probability density function (PDF) for nearest neighbor CTRWs in terms of the input waiting time PDFs. These solutions are also the Laplace space solutions of the generalized master equation (GME). Moreover, based on our counting technique, we introduce the adaptor function for expressing higher order propagators (joint PDFs of time-position variables) for CTRWs in terms of Green's functions. Using the derived formulae, an escape problem from a biased chain is considered.