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Colloquium – Joseph Feneuil, Temple University

January 27 @ 11:00 am - 12:00 pm

TopicThe harmonic measure on low dimensional sets

Abstract: Given a bounded domain $\Omega \subset \R^n$, the harmonic measure with pole in $X$ is a probability measure on the boundary $\partial \Omega$ that satisfies the following property. If $E \sunset \partial \Omega$, then $\omega^X(E)$ is the probability that a particle issued from X and subject to Brownian motion leaves the domain $\Omega$ through $E$. The harmonic measure is strongly related to elliptic PDE, indeed the function on $\Omega$ defined as $u_E(X) = \omega^X(E)$ is the solution to $\Delta u_E = 0$ in $\Omega$, $u_E = 1$ on $E$, $u_E = 0$ on $\partial \Omega \setminus E$. For decades, people tried to figure out what is the geometrical condition on $\partial \Omega$ so that the harmonic measure $\omega^X$ behave like the surface measure – more precisely when the harmonic measure and the surface measure on $\partial \Omega$ are mutually absolutely continuous; and it was ultimately found that under some conditions of topology, the harmonic measure and the surface measure are absolutely continuous with each other (in a quantitative way) if and only if $\partial \Omega$ is uniformly rectifiable (i.e. rectifiable in a quantitative way).

The above criterion is a nice characterization of rectifiable sets using the harmonic measure. However, the harmonic measure is defined only on sets of dimension $n-1$ (a boundary of a domain), which stops {\em a priori} the criterion to be extended to rectifiable sets with lower dimensions (like for instance a line in $\mathbb R^3$). Together with Guy David and Svitlana Mayboroda, we constructed an analogue of the harmonic measure for sets with low dimensions. In this talk, I will explain why the classical harmonic measure cannot be used for lower dimensional sets, I will present our analogue and why we believe it is the right substitute. In particular, we proved that if $E \sunset \mathbb R^n$ is a $d$-dimensional rectifiable set with $d < n-1$, then the harmonic measure that we built is absolutely continuous with respect to the $d$-dimensional Hausdorff measure.


January 27
11:00 am - 12:00 pm
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David Cruz-Uribe