Title: Large-time behavior of two families of operators related to the fractional Laplacian on certain Riemannian manifolds Show Chinese title

Title: 两个与分数拉普拉斯相关的算子族在某些黎曼流形上的长时间行为

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. Show Chinese abstract

Abstract: 本注释关注与分数拉普拉斯算子相关的两个算子族，第一个来自Caffarelli-Silvestre扩展问题，第二个来自分数热方程。它们都包括泊松半群。我们证明，在一个完备、连通且非紧的具有非负里奇曲率的黎曼流形上，两种情况下，带有$L^1$初始数据的解在长时间内渐近地表现为质量乘以基本解。在满足热核Li-Yau双向估计的更一般的流形上，类似的长时间收敛结果仍然有效。在双曲空间上，以及更一般地在单峰非紧对称空间上，情况发生了巨大变化：我们证明对于泊松半群，收敛到泊松核不成立——但在径向初始数据的额外假设下，收敛仍然成立。