Continuous time random walk models for fractional space-time diffusion equations

TL;DR

This paper develops continuous time random walk models that approximate fractional space-time diffusion equations, demonstrating their convergence to time-changed processes via a novel analytic approach.

Contribution

It introduces a new analytic method to prove the convergence of CTRWs to fractional diffusion processes involving stable subordinators.

Findings

Established convergence of CTRWs to fractional diffusion processes.

Provided a new analytic framework for analyzing space-time fractional models.

Enhanced understanding of stochastic processes related to fractional PDEs.

Abstract

In this paper continuous time random walk models approximating fractional space-time diffusion processes are studied. Stochastic processes associated with the considered equations represent time-changed processes, where the time-change process is a L\'evy's stable subordinator with the stability index $\beta \in (0,1).$ In the parer the convergence of constructed CTRWs to time-changed processes associated with the corresponding fractional diffusion equations are proved using a new analytic method.