Let $K$ be a compact connected infinite subset of the plane. Such a set carries a natural measure called harmonic measure which is the hitting distribution of Brownian motion started at $\infty$. This talk will be concerned with multifractal spectrum of this measure or, equivalently according to Frisch-Parisi pionnering work, to the integral-means spectrum of the associated Riemann map. I will focus on deterministic or random cases for which this spectrum may be computed explicitly.
Based on joint work with B. Duplantier (Paris-Saclay), Nguyen Thi Phuong Chi (Ho Chi Minh City) and Han Yong (Shenzen).