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

GOMEZ, IVANA; BELTRITTI, GASTÓN; AIMAR, HUGO

JOURNAL D4ANALYSE MATHEMATIQUE

SPRINGER

Lugar: Berlin; Año: 2018 vol. 136 p. 83 - 101

0021-7670

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 ${tgeq 0}$. 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.