Title: Solvability of the Dirichlet Problem with L^p Data for Caloric Measure

Abstract: This talk concerns two probability measures. First, we consider harmonic measure, which gives solutions to the Dirichlet problem associated to Laplace’s equation. Additionally, we may view harmonic measure as the “hitting probability” for Brownian motion. This probabilistic interpretation shows the connection between the geometry of the domain and properties of harmonic measure.

Next, we discuss caloric measure, which provides a solution to the Dirichlet problem associated to the heat equation. We discuss recent results in the caloric setting, where we aim to prove connections between the geometry of the domain and properties of caloric measure. In particular, we discuss the equivalence of weak-A_infty of caloric measure (with respect to a natural version of “surface measure” on the boundary) and the solvability of the Dirichlet problem with data in L^p for very general domains in space-time. This is joint work with S. Hofmann.