Expected Stopping Time for Lévy Processes on Manifolds

Loosely speaking, a Lévy process looks like Brownian motion interlaced with random jumps at random times. In this expository talk, we will discuss how long it should take a Lévy process to find a small target on a manifold (including two fan favorites: the torus and the sphere) as the target shrinks to a point. To accomplish this, we will study the infinitesimal generator of the Lévy process, with a particular eye toward the impact of geometry and microlocal analysis.