For a domain $Omegasubseteqmathbb{R}^{d+1}$, and some boundary data $fin C(partialOmega)$, we can (usually) solve the Dirichlet problem and find a harmonic function $u_{f}$ that agrees with $f$ on $partialOmega$. For $x_{0}in Omega$, the association $frightarrow u_{f}(x_{0})$. is a linear functional, so the Riesz Representation gives us a measure $omega_{Omega}^{x_{0}}$ on $partialOmega$ called the harmonic measure with pole at $x_{0}$. One can also think of the harmonic measure of a set $Esubseteq partialOmega$ as the probability that a Brownian motion of starting at $x_{0}$ will first hit the boundary in $E$.
It is an interesting problem to establish the relationship between the measure theoretic behavior of a domain’s harmonic measure and the geometry of the boundary. For example, in the plane, a classical result of the Riesz brothers says that the harmonic measure is absolutely continuous with respect to 1-dimensional Hausdorff measure (or length measure) if and only if the boundary is a rectifiable curve (that is, the boundary can be parametrized by a Lipschitz map). If the harmonic measures from the interior and exterior of a Jordan domain are mutually absolutely continuous, this implies both measures can be covered up to measure zero by Lipschitz curves by a result of Bishop, Carleson, Garnett, and Jones. Establishing analogous results in higher dimensions is more difficult, as techniques from complex analysis are no longer available, and certain pathological examples begin to appear.
In the first half of this talk, we will introduce the audience to the notion of harmonic measure, rectifiable sets and purely unrectifiable sets, and survey some very recent results about the connections between absolute continuity of harmonic measure and rectifiability of the boundary in dimensions larger than two. In the second half, we will go over some of the various techniques from different fields used in proving them, such as Preiss’ theory of tangent measures from geometric measure theory, Green’s functions with poles at infinity used by Kenig and Toro, and the theory of singular integral operators on subsets of Euclidean space.
The results we present are joint works with Murat Akman, Steve Hofmann, José María Martell, Svitlana Mayboroda, Mihalis Mourgoglou, Xavier Tolsa, and Alexander Volberg.