Continuous time random walk as a random walk in a random environment

TL;DR

This paper demonstrates that certain continuous time random walks can be represented as random walks in random environments, with convergence properties linked to stable laws and renewal processes, providing new insights into their limiting behavior.

Contribution

It introduces a novel representation of CTRWs as random walks in random environments for renewal times in a dense subset of the domain of attraction of stable laws.

Findings

Established quenched limits for CTRWs in random environments

Provided bounds on approximation errors for these models

Linked renewal processes to stable law convergence

Abstract

We show that for a weakly dense subset of the domain of attraction of a positive stable random variable of index $0<\alpha<1$($DOA\left(\alpha\right))$ the functional stable convergence is a time-changed renewal convergence of distribution of finite mean. Applied to Continuous Time Random Walk(CTRW) \'a la Montroll and Wiess we show that CTRW with renewal times in a weakly dense set of $DOA\left(\alpha\right)$ can be realized as random walk in a random environment. We find the quenched limit and give a bound on the error of the approximation.