A nonlocal diffusion problem on manifolds

TL;DR

This paper investigates a nonlocal diffusion equation on manifolds, establishing existence, uniqueness, and convergence to classical operators, while analyzing long-term behavior and special cases like hyperbolic space.

Contribution

It introduces a new analysis of nonlocal diffusion on manifolds, including convergence results and spectral analysis, extending Euclidean results to curved spaces.

Findings

Solutions exist and are unique.

Operator converges to the Heat-Beltrami operator under rescaling.

Distinct long-term behavior in hyperbolic space.

Abstract

In this paper we study a nonlocal diffusion problem on a manifold. These kind of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. We first prove existence and uniqueness of solutions and a comparison principle. Then, for a convenient rescaling we prove that the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior on compact manifolds by studying the spectral properties of the operator. Finally, for the model case of hyperbolic space we study the long time asymptotics and find a different and interesting behavior.