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 $$\chi _{A_n} (x) \rightarrow X(x)$$ weakly in $$L^\infty $$ (and then $$\chi _{B_n} (x) \rightarrow 1-X(x)$$ weakly in $$L^\infty $$ ) as $$n \rightarrow \infty $$ 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.