Abstract or Additional Information
Given a domain, the (Harmonic/Caloric) Green Function and Elliptic/Caloric measure are connected through the so-called Riesz formula. This connection allows one to connect the solution map for the Dirichlet boundary value problem, that is, integration against the caloric/harmonic measure, with an object that lives in the interior of the domain (the Green function). Through a variety of methods one can show that the caloric/elliptic measure is `nice’ if the Green function acts like the distance to the boundary `most/all of the time’.
Perhaps more surprising is how the behavior of the Green function is capable of detecting the geometry of the domain. As an example, the property that the `Green function acts like the distance to the boundary most/all of the time’ implies that the boundary is in some sense flat. Of course, the nature of this flatness depends on how exactly we state that the `Green function acts like the distance to the boundary most/all of the time’.
I will discuss the history and developments concerning these connections and conclude with some of my recent and future work with Hofmann, Nyström and Martell.