Continuous time random walks and the Cauchy problem for the heat equation

TL;DR

This paper investigates anomalous diffusions modeled by continuous time random walks, establishing their connection to the heat equation through scaling limits and providing a framework for solving related Cauchy problems.

Contribution

It introduces a method to analyze CTRW-induced diffusions, linking them to classical heat equations via weak convergence under parabolic rescaling.

Findings

Solutions tend to the heat equation under scaling

Weak convergence of CTRW models to classical diffusion

Framework for solving CTRW-based Cauchy problems

Abstract

In this paper we deal with anomalous diffusions induced by Continuous Time Random Walks - CTRW in $\mathbb{R}^n$. A particle moves in $\mathbb{R}^n$ in such a way that the probability density function $u(\cdot,t)$ of finding it in region $\Omega$ of $\mathbb{R}^n$ is given by $\int_{\Omega}u(x,t) dx$. The dynamics of the diffusion is provided by a space time probability density $J(x,t)$ compactly supported in $\{t\geq 0\}$. For $t$ large enough, $u$ must satisfy the equation $u(x,t)=[(J-\delta)\ast u](x,t)$ where $\delta$ is the Dirac delta in space time. We give a sense to a Cauchy type problem for a given initial density distribution $f$. We use Banach fixed point method to solve it, and we prove that under parabolic rescaling of $J$ the equation tends weakly to the heat equation and that for particular kernels $J$ the solutions tend to the corresponding temperatures when the scaling parameter approaches to zero.