A subharmonicity property of harmonic measures

Recently it has been established that for compact sets F lying on a circle S, the harmonic measure in the complement of F with respect to any point a ∈ S \ F has convex density on any arc of F . In this note we give an alternative proof of this fact which is based on random walks, and which also yields an analogue in higher dimensions: for compact sets F lying on a sphere S in Rn, the harmonic measure in the complement of F with respect to any point a ∈ S \F is subharmonic in the interior of F . 1 The result in two dimensions Let G be a domain G ⊂ R with compact boundary. In what follows, we denote the n-dimensional harmonic measure for a point z ∈ G by ω(·, z, G) (if it exists). So this is a measure on the boundary of G and it is the reproducing measure for harmonic functions in G: if u is harmonic in G (including at infinity if G is unbounded) and continuous on the closure G, then