Inmathematics, especiallypotential theory,harmonic measure is a concept related to the theory ofharmonic functions that arises from the solution of the classicalDirichlet problem.

Inprobability theory, the harmonic measure of a subset of the boundary of a bounded domain inEuclidean space${\displaystyle R^n}$,${\displaystyle n\ge 2}$ is the probability that aBrownian motion started inside a domain hits that subset of the boundary. More generally, harmonic measure of anItō diffusionX describes the distribution ofX as it hits the boundary ofD. In thecomplex plane, harmonic measure can be used to estimate themodulus of ananalytic function inside a domainD given bounds on the modulus on theboundary of the domain; a special case of this principle isHadamard's three-circle theorem. On simply connected planar domains, there is a close connection between harmonic measure and the theory ofconformal maps.

The termharmonic measure was introduced byRolf 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]

LetD be abounded,open domain inn-dimensionalEuclidean spaceRⁿ,n ≥ 2, and let ∂D denote the boundary ofD. Anycontinuous functionf : ∂D → R determines a uniqueharmonic functionH_f that solves theDirichlet problem

If a pointx ∈ D is fixed, by theRiesz–Markov–Kakutani representation theorem and themaximum principleH_f(x) determines aprobability measureω(x, D) on ∂D by

${\displaystyle H_f(x)={\int }_{\mathrm{\partial }D}f(y)\mathrm{d}\omega (x,D)(y).}$

The measureω(x, D) is called theharmonic measure (of the domainD with pole atx).

• For any Borel subsetE of ∂D, the harmonic measureω(x, D)(E) is equal to the value atx of the solution to the Dirichlet problem with boundary data equal to theindicator function ofE.
• For fixedD andE ⊆ ∂D,ω(x, D)(E) is a harmonic function ofx ∈ D and

${\displaystyle 0\le \omega (x,D)(E)\le 1;}$

${\displaystyle 1-\omega (x,D)(E)=\omega (x,D)(\mathrm{\partial }D\setminus E);}$

Hence, for eachx andD,ω(x, D) is aprobability measure on ∂D.

• Ifω(x, D)(E) = 0 at even a single pointx ofD, then${\displaystyle y\mapsto \omega (y,D)(E)}$ is identically zero, in which caseE is said to be a set ofharmonic measure zero. This is a consequence ofHarnack's inequality.

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

• F. and M. Riesz Theorem:^[4] If${\displaystyle D\subset {\mathbb{R}}^2}$ is a simply connected planar domain bounded by arectifiable curve (i.e. if), then harmonic measure is mutually absolutely continuous with respect to arc length: for all${\displaystyle E\subset \mathrm{\partial }D}$,${\displaystyle \omega (X,D)(E)=0}$ if and only if${\displaystyle H^1(E)=0}$.
• Makarov's theorem:^[5] Let${\displaystyle D\subset {\mathbb{R}}^2}$ be a simply connected planar domain. If${\displaystyle E\subset \mathrm{\partial }D}$ and${\displaystyle H^s(E)=0}$ for some, then${\displaystyle \omega (x,D)(E)=0}$. Moreover, harmonic measure onD ismutually singular with respect tot-dimensional Hausdorff measure for all t > 1.

• Dahlberg's theorem:^[6] If${\displaystyle D\subset {\mathbb{R}}^n}$ is a boundedLipschitz domain, then harmonic measure and (n − 1)-dimensional Hausdorff measure are mutually absolutely continuous: for all${\displaystyle E\subset \mathrm{\partial }D}$,${\displaystyle \omega (X,D)(E)=0}$ if and only if${\displaystyle H^{n-1}(E)=0}$.

• If is the unit disk, then harmonic measure of${\displaystyle \mathbb{D}}$ with pole at the origin is length measure on the unit circle normalized to be a probability, i.e.${\displaystyle \omega (0,\mathbb{D})(E)=|E|/2\pi }$ for all${\displaystyle E\subset S^1}$ where${\displaystyle |E|}$ denotes the length of${\displaystyle E}$.
• If${\displaystyle \mathbb{D}}$ is the unit disk and${\displaystyle X\in \mathbb{D}}$, then${\displaystyle \omega (X,\mathbb{D})(E)={\int }_E\frac{1-|X{|}^2}{|X-Q{|}^2}\frac{d{H}^1(Q)}{2\pi }}$ for all${\displaystyle E\subset S^1}$ where${\displaystyle H^1}$ denotes length measure on the unit circle. TheRadon–Nikodym derivative${\displaystyle d\omega (X,\mathbb{D})/d{H}^1}$ is called thePoisson kernel.
• More generally, if${\displaystyle n\ge 2}$ and is then-dimensional unit ball, then harmonic measure with pole at${\displaystyle X\in {\mathbb{B}}^n}$ is${\displaystyle \omega (X,{\mathbb{B}}^n)(E)={\int }_E\frac{1-|X{|}^2}{|X-Q{|}^n}\frac{d{H}^{n-1}(Q)}{{\sigma }_{n-1}}}$ for all${\displaystyle E\subset S^{n-1}}$ where${\displaystyle H^{n-1}}$ denotes surface measure ((n − 1)-dimensionalHausdorff measure) on the unit sphere${\displaystyle S^{n-1}}$ and${\displaystyle H^{n-1}(S^{n-1})={\sigma }_{n-1}}$.

• 

  If${\displaystyle D\subset {\mathbb{R}}^2}$ is a simply connected planar domain bounded by aJordan curve andX${\displaystyle \in }$D, then${\displaystyle \omega (X,D)(E)=|f^{-1}(E)|/2\pi }$ for all${\displaystyle E\subset \mathrm{\partial }D}$ where${\displaystyle f:\mathbb{D}\to D}$ is the uniqueRiemann map which sends the origin toX, i.e.${\displaystyle f(0)=X}$. SeeCarathéodory's theorem.
• If${\displaystyle D\subset {\mathbb{R}}^2}$ is the domain bounded by theKoch snowflake, then there exists a subset${\displaystyle E\subset \mathrm{\partial }D}$ of the Koch snowflake such that${\displaystyle E}$ has zero length (${\displaystyle H^1(E)=0}$) and full harmonic measure${\displaystyle \omega (X,D)(E)=1}$.

Consider anRⁿ-valued Itō diffusionX starting at some pointx in the interior of a domainD, with lawP^x. Suppose that one wishes to know the distribution of the points at whichX exitsD. For example, canonical Brownian motionB on thereal line starting at 0 exits theinterval (−1, +1) at −1 with probability⁠1/2⁠ and at +1 with probability⁠1/2⁠, soB_{τ(−1, +1)} isuniformly distributed on the set {−1, +1}.

In general, ifG iscompactly embedded withinRⁿ, then theharmonic measure (orhitting distribution) ofX on the boundary ∂G ofG is the measureμ_G^x defined by

${\displaystyle {\mu }_G^x(F)={\mathbf{P}}^x[X_{{\tau }_G}\in F]}$

forx ∈ G andF ⊆ ∂G.

Returning to the earlier example of Brownian motion, one can show that ifB is a Brownian motion inRⁿ starting atx ∈ Rⁿ andD ⊂ Rⁿ is anopen ball centred onx, then the harmonic measure ofB on ∂D isinvariant under allrotations ofD aboutx and coincides with the normalizedsurface measure on ∂D

• Garnett, John B.; Marshall, Donald E. (2005).Harmonic Measure. Cambridge: Cambridge University Press.ISBN 978-0-521-47018-6.

• Øksendal, Bernt K. (2003).Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer.ISBN 3-540-04758-1.MR 2001996 (See Sections 7, 8 and 9)

• Capogna, Luca; Kenig, Carlos E.;Lanzani, Loredana (2005).Harmonic Measure: Geometric and Analytic Points of View. University Lecture Series. Vol. ULECT/35. American Mathematical Society. p. 155.ISBN 978-0-8218-2728-4.

• P. Jones and T. Wolff, Hausdorff dimension of Harmonic Measure in the plane, Acta. Math. 161 (1988) 131-144 (MR962097)(90j:31001)
• C. Kenig and T. Toro, Free Boundary regularity for Harmonic Measores and Poisson Kernels, Ann. of Math. 150 (1999)369-454MR 172669992001d:31004)
• C. Kenig, D. Preissand, T. Toro, Boundary Structure and Size in terms of Interior and Exterior Harmonic Measures in Higher Dimensions, Jour. of Amer. Math. Soc. vol 22 July 2009, no3,771-796
• S. G. Krantz, The Theory and Practice of Conformal Geometry, Dover Publ. Mineola New York (2016) esp. Ch 6 classical case

• Solomentsev, E.D. (2001) [1994],"Harmonic measure",Encyclopedia of Mathematics,EMS Press

• Measures (measure theory)
• Potential theory

• CS1: long volume value