Student Harmonic Analysis and PDE Seminar (HADES): Harmonic measures in the plane and higher-dimensional spaces

Seminar | March 21 | 3:40-5 p.m. | 740 Evans Hall

 Zihui Zhao, University of Washington

 Department of Mathematics

Given a domain Ω, consider a Brownian motion starting from an interior point $x_0$, the harmonic measure $\omega =\omega ^{x_0}$ describes where the Brownian motion exits from the domain. Information of harmonic functions is also encoded in \omega : given a continuous function $f \in C(\partial \Omega )$, let $u$ be its harmonic extension in Ω, we have $u(x_0) = \int f d\omega $. A natural question arises: how is ω related to the surface measure of the boundary? In other words, does the Brownian traveler 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.