Homogenization for nonlocal evolution problems with three different smooth kernels

Abstract

In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain is divided into a sequence of two subdomains $A_n\cup B_n$ and we have three different smooth kernels, one that controls the jumps from $A_n$ to $A_n$, a second one that controls the jumps from $B_n$ to $B_n$ and the third one that governs the interactions between $A_n$ and $B_n$.Assuming that ${\chi }_{A_n}(x)\to X(x)$ weakly in $L^{\infty }$ (and then ${\chi }_{B_n}(x)\to 1-X(x)$ weakly in $L^{\infty }$) as $n\to \infty$ and that the initial condition is given by a density $u_0$ in $L^2$ we show that there is an homogenized limit system in which the three kernels and the limit function $X$ appear. When the initial condition is a delta at one point, ${\delta }_{\bar{x}}$ (this corresponds to the process that starts at $\bar{x}$) we show that there is convergence along subsequences such that $\bar{x}\in A_{n_j}$ or $\bar{x}\in B_{n_j}$ for every $n_j$ large enough. We also provide a probabilistic interpretation of this evolution equation in terms of a stochastic process that describes the movement of a particle that jumps in $\Omega$ according to the three different kernels and show that the underlying process converges in distribution to a limit process associated with the limit equation. We focus our analysis in Neumann type boundary conditions and briefly describe at the end how to deal with Dirichlet boundary conditions.

arXiv admin note: text overlap with arXiv:2003.03407