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. In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space R n {\displaystyle R^{n}} , n ≥ 2 {\displaystyle n\geq 2} 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, although Nevanlinna notes the idea appeared implicitly in earlier work by Johansson, Frigyes Riesz, Marcel Riesz, Torsten Carleman, Alexander Ostrowski and Gaston Julia. The connection between harmonic measure and Brownian motion was first identified by Kakutani in 1944.
Definition 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
{ − Δ H f ( x ) = 0 , x ∈ D ; H f ( x ) = f ( x ) , x ∈ ∂ D . {\displaystyle {\begin{cases}-\Delta H_{f}(x)=0,&x\in D;\\H_{f}(x)=f(x),&x\in \partial D.\end{cases}}}
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
H f ( x ) = ∫ ∂ D f ( y ) d ω ( x , D ) ( y ) . {\displaystyle H_{f}(x)=\int _{\partial D}f(y)\,\mathrm {d} \omega (x,D)(y).}
The measure ω(x, D) is called the harmonic measure (of the domain D with pole at x).
Properties For any Borel subset E of ∂D, the harmonic measure ω(x, D)(E) 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(x, D)(E) is a harmonic function of x ∈ D and
0 ≤ ω ( x , D ) ( E ) ≤ 1 ; {\displaystyle 0\leq \omega (x,D)(E)\leq 1;}
1 − ω ( x , D ) ( E ) = ω ( x , D ) ( ∂ D ∖ E ) ; {\displaystyle 1-\omega (x,D)(E)=\omega (x,D)(\partial D\setminus E);}
Hence, for each x and D, ω(x, D) is a probability measure on ∂D. If ω(x, D)(E) = 0 at even a single point x of D, then y ↦ ω ( y , D ) ( E ) {\displaystyle y\mapsto \omega (y,D)(E)} is identically zero, in which case E is said to be a set of harmonic measure zero. This is a consequence of Harnack'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.
… excerpt ends here. Continue reading the full article.
