Mixing local and nonlocal evolution equations

Abstract: In this paper we study the homogenization of a stochastic process and its associated evolution equations in which we mix a local part (given by a Brownian motion with a reflection on the boundary) and a nonlocal part (given by a jump process with a smooth kernel). We consider a sequence of partitions of the (fixed) spacial domain into two parts (local and nonlocal) that are mixed in such a way that they both have positive density at every point in the limit. Under adequate hypotheses on the sequence of partitions, we prove convergence of the associated densities (that are solutions to an evolution equation with coupled local and nonlocal parts in two different regions of the domain) to the unique solution to a limit evolution system in which the local part disappears and the nonlocal part survives but divided into two different components. We also obtain convergence in distributions of the processes associated to the partitions and prove that the limit process has a density pair that coincides with the limit of the densities.