Analysis Seminar

Topic: Elliptic measures and the geometry of domains

Speaker: Zihui Zhao

Affiliation: Member, School of Mathematics

Date & Time: Thursday February 14th, 2019, 1:00pm - 2:00pm

Location: Simonyi Hall 101

Video: 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 $\sigma$.