Lagging/Leading Coupled Continuous Time Random Walks, Renewal Times and their Joint Limits

TL;DR

This paper investigates the limiting behavior of coupled continuous time random walks with power law waiting times and jumps, identifying four joint limit processes and their laws, which advance understanding of anomalous diffusion models.

Contribution

It introduces a detailed analysis of the joint limits of CTRWs with coupled waiting times and jumps, including renewal times, providing explicit joint laws for the limiting processes.

Findings

Identified two different limit processes depending on the order of waiting times and jumps.

Derived the joint law of four limit processes at a fixed time.

Extended the understanding of CTRW limits with renewal times.

Abstract

Subordinating a random walk to a renewal process yields a continuous time random walk (CTRW) model for diffusion, including the possibility of anomalous diffusion. Transition densities of scaling limits of power law CTRWs have been shown to solve fractional Fokker-Planck equations. We consider limits of sequences of CTRWs which arise when both waiting times and jumps are taken from an infinitesimal triangular array. We identify two different limit processes $X_t$ and $Y_t$ when waiting times precede or follow jumps, respectively. In the limiting procedure, we keep track of the renewal times of the CTRWs and hence find two more limit processes. Finally, we calculate the joint law of all four limit processes evaluated at a fixed time $t$.