Absolute continuity and rectifiability of harmonic measure

Jonas Azzam (University of Edinburgh)

Frank Adams 1,

Given a domain \(\Omega\subseteq \mathbb{R}^{d+1}\), the harmonic measure of a subset \(A\) of the boundary is the probability that the Brownian motion starting at some fixed point in the domain first exists \(\Omega\) through \(A\). It is a popular problem to study how the behavior of the harmonic measure influences the geometry of the boundary and vice versa. For example, what happens when two domains share a boundary (i.e. they are complementary) and their harmonic measures are mutually absolutely continuous? In this talk, I will discuss the background and solution to this problem, which builds off of the tangent measure techniques developed by Kenig, Preiss, and Toro (which I will emphasize more) and uses some new techniques from singular integrals developed by Girela-Sarrión and Tolsa. This is joint work with Mourgoglou, Tolsa, and Volberg. 

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