# Harmonic measures in the plane and higher-dimensional spaces

Zihui Zhao
Thursday, October 19, 2017 - 2:30pm to 3:20pm
PDL C-401

Abstract: Given a domain $\dpi{300}\inline \Omega$, consider a Brownian motion starting from an interior point $\dpi{300}\inline x_0$, the harmonic measure $\dpi{300}\inline \omega=\omega^{x_0}$ describes where the Brownian motion exits from the domain. Information of harmonic functions is also encoded in $\dpi{300}\inline \omega$: given a continuous function $\dpi{300}\inline f\in C(\partial\Omega)$, let $\dpi{300}\inline u$ be its harmonic extension in $\dpi{300}\inline \Omega$, we have $\dpi{300}\inline u(x_0) = \int f d\omega$. A natural question arises: how is $\dpi{300}\inline \omega$ related to the surface measure of the boundary? In other words, does the Brownian traveller see things differently from our naked eyes? The answer to this question in turn reveals geometric information of the domain. This talk will review some cornerstone results in this topic for the plane and higher dimensions.

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