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Analysis Seminar

Thursday, November 1, 2018 - 3:00pm

Zihui Zhao



University of Pennsylvania


Given a bounded domain $\Omega$, the harmonic measure $\omega$ is a probability measure on $\partial \Omega$ and it characterizes where a Brownian traveler in $\Omega$ is likely to exit the domain from $\partial \Omega$. The elliptic measure is a non-homogeneous variant of harmonic measure.
Since 1917, there has been much study about the relationship between the elliptic/harmonic 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 $\sigma$.