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.