Semi-Markov approach to continuous time random walk limit processes

TL;DR

This paper introduces a semi-Markov framework for analyzing continuous time random walk (CTRW) limit processes, enabling explicit computation of their finite-dimensional distributions and handling complex jump-waiting time couplings.

Contribution

It develops a general semi-Markov theory for CTRW limits, embedding non-Markovian processes into Markov processes via state space augmentation.

Findings

Explicit transition kernels derived for CTRW limit processes

Method allows computation of all finite-dimensional distributions

Two illustrative examples demonstrate the approach

Abstract

Continuous time random walks (CTRWs) are versatile models for anomalous diffusion processes that have found widespread application in the quantitative sciences. Their scaling limits are typically non-Markovian, and the computation of their finite-dimensional distributions is an important open problem. This paper develops a general semi-Markov theory for CTRW limit processes in $\mathbb{R}^d$ with infinitely many particle jumps (renewals) in finite time intervals. The particle jumps and waiting times can be coupled and vary with space and time. By augmenting the state space to include the scaling limits of renewal times, a CTRW limit process can be embedded in a Markov process. Explicit analytic expressions for the transition kernels of these Markov processes are then derived, which allow the computation of all finite dimensional distributions for CTRW limits. Two examples illustrate the proposed method.