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

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