In mathematics, especially potential theory, harmonic measure is a concept related to the theory ofharmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space, is the probability that a Brownian motion started inside a domain hits that subset of the boundary. More generally, harmonic measure of an Itō diffusionX 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. On simply connected planar domains, there is a close connection between harmonic measure and the theory of conformal maps. The term harmonic measure was introduced by Rolf Nevanlinna in 1928 for planar domains, although Nevanlinna notes the idea appeared implicitly in earlier work by Johansson, F. Riesz, M. Riesz, Carleman, Ostrowski and Julia. The connection between harmonic measure and Brownian motion was first identified by Kakutani ten years later in 1944.
For any Borel subsetE of ∂D, the harmonic measure ω is equal to the value at x of the solution to the Dirichlet problem with boundary data equal to the indicator function of E.
For fixed D and E ⊆ ∂D, ω is an harmonic function of x ∈ D and
Since explicit formulas for harmonic measure are not typically available, we are interested in determining conditions which guarantee a set has harmonic measure zero.
Makarov's theorem: Let be a simply connected planar domain. If and for some, then. Moreover, harmonic measure on D is mutually singular with respect to t-dimensional Hausdorff measure for all t > 1.
Dahlberg's theorem: If is a bounded Lipschitz domain, then harmonic measure and -dimensional Hausdorff measure are mutually absolutely continuous: for all, if and only if.
Examples
If is the unit disk, then harmonic measure of with pole at the origin is length measure on the unit circle normalized to be a probability, i.e. for all where denotes the length of.
More generally, if and is the n-dimensional unit ball, then harmonic measure with pole at is for all where denotes surface measure on the unit sphere and.
If is a simply connected planar domain bounded by a Jordan curve and XD, then for all where is the unique Riemann map which sends the origin to X, i.e.. See Carathéodory's theorem.
If is the domain bounded by the Koch snowflake, then there exists a subset of the Koch snowflake such that has zero length and full harmonic measure.
The harmonic measure of a diffusion
Consider an Rn-valued Itō diffusion X starting at some point x in the interior of a domain D, with law Px. Suppose that one wishes to know the distribution of the points at which X exits D. For example, canonical Brownian motion B on the real line starting at 0 exits the interval at −1 with probability ½ and at +1 with probability ½, so Bτ is uniformly distributed on the set. In general, if G is compactly embedded within Rn, then the harmonic measure of X on the boundary ∂G of G is the measure μGx defined by for x ∈ G and F ⊆ ∂G. Returning to the earlier example of Brownian motion, one can show that if B is a Brownian motion in Rn starting at x ∈ Rn and D ⊂ Rn is an open ball centred on x, then the harmonic measure of B on ∂D is invariant under all rotations of D about x and coincides with the normalized surface measure on ∂D
