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Elliptic measures and the geometry of the domains

Zihui Zhao, University of Washington
Tuesday, May 1, 2018 - 1:30pm to 3:30pm
PDL C-401
Given a bounded domain \Omega, the harmonic measure \omega is a probability measure on \partial\Omegaand it characterizes where a Brownian traveller 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 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.