Elliptic measures and the geometry of domains

Analysis Seminar
Topic:Elliptic measures and the geometry of domains
Speaker:Zihui Zhao
Affiliation:Member, School of Mathematics
Date:Thursday, February 14
Time/Room:1:00pm - 2:00pm/Simonyi Hall 101
Video Link:https://video.ias.edu/analysis/2019/0214-ZihuiZhao

Given a bounded domain $\Omega$, the harmonic measure $\omega$ is a probability measure on $\partial \Omega$ and it characterizes where a Brownian traveller moving in $\Omega$ is likely to exit the domain from. The elliptic measure is a non-homogenous variant of harmonic measure. Since 1917, there has been much study about the relationship between the harmonic/elliptic measure $\omega$ and the surface measure $\sigma$ of the boundary. In particular, are $\omega$ and $\sigma$ absolutely continuous with each other? In this talk, I will show how a positive answer to this question implies that the corresponding domain enjoys good geometric property, thus we obtain a sufficient condition for the absolute continuity of $\omega$ and $\sigma$.