In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem.

Harmonic measure is the exit distribution of Brownian motion

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 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. 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,[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(xD) on ∂D by

The measure ω(xD) is called the harmonic measure (of the domain D with pole at x).


Hence, for each x and D, ω(xD) is a probability measure on ∂D.

Since explicit formulas for harmonic measure are not typically available, we are interested in determining conditions which guarantee a set has harmonic measure zero.


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 (−1, +1) at −1 with probability ½ and at +1 with probability ½, so Bτ(−1, +1) is uniformly distributed on the set {−1, +1}.

In general, if G is compactly embedded within Rn, then the harmonic measure (or hitting distribution) 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

General references


  1. ^ R. Nevanlinna (1970), "Analytic Functions", Springer-Verlag, Berlin, Heidelberg, cf. Introduction p. 3
  2. ^ R. Nevanlinna (1934), "Das harmonische Mass von Punktmengen und seine Anwendung in der Funktionentheorie", Comptes rendus du huitème congrès des mathématiciens scandinaves, Stockholm, pp. 116–133.
  3. ^ Kakutani, S. (1944). "On Brownian motion in n-space". Proc. Imp. Acad. Tokyo. 20 (9): 648–652. doi:10.3792/pia/1195572742.
  4. ^ F. and M. Riesz (1916), "Über die Randwerte einer analytischen Funktion", Quatrième Congrès des Mathématiciens Scandinaves, Stockholm, pp. 27–44.
  5. ^ Makarov, N. G. (1985). "On the Distortion of Boundary Sets Under Conformal Maps". Proc. London Math. Soc. 3. 52 (2): 369–384. doi:10.1112/plms/s3-51.2.369.
  6. ^ Dahlberg, Björn E. J. (1977). "Estimates of harmonic measure". Arch. Rat. Mech. Anal. 65 (3): 275–288. Bibcode:1977ArRMA..65..275D. doi:10.1007/BF00280445. S2CID 120614580.