Abstract: In 1977 Dahlberg resolved a long-standing question concerning mutual absolute continuity of harmonic measure and surface measure in Lipschitz domains. He, in fact, showed that the harmonic measure satisfies a certain quantitative version of (mutual) absolute continuity known as the \(A_\infty\) condition. It was later conjectured by Hunt that the most natural parabolic analogue holds, that is, if Omega is the region above a parabolic Lipschitz (Lip(1,1/2)) graph, then the caloric measure satisfies a (parabolic) \(A_\infty\) condition. Kaufman and Wu disproved this conjecture, but it was shown by Lewis and Murray that if the function defining the graph domain has the additional property that it has a 1/2-order time derivative in (parabolic) BMO then the caloric measure does satisfy a (parabolic) \(A_\infty\) condition.
In my recent joint work with Hofmann, Martell and Nyström we have shown that the condition of Lewis and Murray is sharp, that is, if Omega is the region above a parabolic Lipschitz graph and the caloric measure is in \(A_\infty\) then the graph function has a 1/2-order time derivative in (parabolic) BMO. This resolves a 30-year-old conjecture. Moreover, it is known that such graphs are “parabolic uniformly rectifiable” and, in forthcoming work, we will show that the (weak) \(A_\infty\) property of caloric measure in a domain implies the boundary is parabolic uniformly rectifiable in vast generality. I will discuss these results in a friendly manner, mostly drawing pictures and giving the general ideas/principles that go into the proofs.