Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> 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.</jats:p>