Taft Lectures

There is a rich and fruitful interplay between analysis and probability theory. Transition density function of a Markov process is the fundamental solution, also called heat kernel, of its infinitesimal generator. In this talk, I will explain how probabilistic insights can help in the study of heat kernels in the context of rectilinear fractional Laplacians. These non-local operators are the infinitesimal generators of rectilinear α-stable processes, which are Levy processes on ℝ^d, whose coordinate processes are independent copies of one-dimensional α-stable processes. They have many distinct properties from that of isotropic fractional Laplacians. I will discuss the geometric characterization of an open subset D so that the Dirichlet heat kernels p_D(t, x, y) on D are strictly positive. I will further present results on the properties of p_D(t, x, y) including its regularity as well as the sharp two-sided bounds on C^1,1-domain.

Refreshments will be served 3:15-3:45pm in the Math Faculty 
& Graduate Student Lounge Room 4118 French Hall West