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 An ∪ Bn and we have three different smooth kernels, one that controls the jumps from An to An, a second one that controls the jumps from Bn to Bn and the third one that governs the interactions between An and Bn.Assuming that χA n (x) → X(x) weakly in L ∞ (and then χB n (x) → 1 − X(x) weakly in L ∞ ) as n → ∞ and that the initial condition is given by a density u0 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, δx (this corresponds to the process that starts at x) we show that there is convergence along subsequences such that x ∈ An j or x ∈ Bn j for every nj 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 Ω 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.