Large-time behavior of two families of operators related to the fractional Laplacian on certain Riemannian manifolds

TL;DR

This paper investigates the long-time behavior of operators related to the fractional Laplacian on various Riemannian manifolds, showing convergence results on some manifolds and failure of convergence on hyperbolic spaces.

Contribution

It establishes asymptotic convergence of fractional Laplacian operators on certain manifolds and highlights the failure of this convergence on hyperbolic spaces, with conditions for radial data.

Findings

Convergence to fundamental solutions on manifolds with non-negative Ricci curvature.

Long-time behavior aligns with the mass times the fundamental solution.

Failure of convergence on hyperbolic space for the Poisson semigroup.

Abstract

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with $L^1$ initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.