Martin Ulmer will give a talk on 02/03/2026.

The harmonic measure describes the probability that a Brownian traveler originating from a fixed point exits a domain through a given set. In 1988, Jones and Wolff showed that if we have a domain subset of $\mathbb{R}^2$, the dimension of the support of the harmonic measure is 1. That means that even if the domain is a fractal with Hausdorff dimension greater than 1 (like the Koch snowflake), the “landing area” of the harmonic measure is just one dimensional. However, for domains in $\mathbb{R}^n$, we know significantly less. Bourgain proved in 1987 a dimension drop, i.e. that the dimension of the harmonic measure is less than $n-\alpha$, and Wolff (1995) constructed a fractal domain for which the harmonic measure has dimension $n+\beta$. Both numbers $\alpha$ and $\beta$ are very small. In this talk we would like to go through Wolff’s snowflake construction in $\mathbb{R}^3$ to understand why the harmonic measure can have a dimension strictly greater than 2.