Spatial maxima, unimodality, and asymptotic behaviour of solutions to discrete diffusion-type equations

Abstract

We consider a class of partial difference equations which generalizes the discrete diffusion (heat) equation. In the first part of the paper, we study their fundamental solutions, and focus on the location of their spatial maxima and unimodality. In the second part, we describe the asymptotic behaviour of solutions with arbitrary bounded initial data. Throughout the paper, we take advantage of the relation between discrete diffusion-type equations and random walks on the set of integers.