UBC Probability Seminar: Mathav Murugan

Spitzer (1958) showed that the trace of reflected Brownian motion on the upper half space on its boundary is a Cauchy process. More generally, every symmetric stable process in n-dimensional Euclidean space can be obtained as a trace process of a diffusion in the (n+1)-dimensional upper half-space. This was observed by Molchanov and Ostrowski (1969) and later rediscovered in a celebrated work of Caffarelli and Silvestre (2007). Our main result suggests that such a stable-like behaviour of the boundary trace of reflected diffusions is a generic phenomenon. In particular, we obtain stable-like heat kernel estimates for the boundary trace process of a reflected diffusion on a large class of domains when the diffusion on the underlying ambient space satisfies sub-Gaussian heat kernel estimates. Our arguments rely on new results of independent interest such as sharp estimates and doubling properties of the harmonic measure, continuous extension of Naïm kernel to the topological boundary, and the Doob-Naïm formula for the energy of the boundary trace process. This is joint work with Naotaka Kajino.