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).