Harmonic measure

In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space

More generally, harmonic measure of an Itō diffusion X describes the distribution of X as it hits the boundary of D. In the complex plane, harmonic measure can be used to estimate the modulus of an analytic function inside a domain D given bounds on the modulus on the boundary of the domain; a special case of this principle is Hadamard's three-circle theorem.

The term harmonic measure was introduced by Rolf Nevanlinna in 1928 for planar domains,[1][2] although Nevanlinna notes the idea appeared implicitly in earlier work by Johansson, F. Riesz, M. Riesz, Carleman, Ostrowski and Julia (original order cited).

The connection between harmonic measure and Brownian motion was first identified by Kakutani ten years later in 1944.

[3] Let D be a bounded, open domain in n-dimensional Euclidean space Rn, n ≥ 2, and let ∂D denote the boundary of D. Any continuous function f : ∂D → R determines a unique harmonic function Hf that solves the Dirichlet problem If a point x ∈ D is fixed, by the Riesz–Markov–Kakutani representation theorem and the maximum principle Hf(x) determines a probability measure ω(x, D) on ∂D by The measure ω(x, D) is called the harmonic measure (of the domain D with pole at x).