Harmonic measure

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In mathematics, harmonic measure is a concept that arises in the theory of harmonic functions, where it 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. In a closely related area, the harmonic measure of an Itō diffusion X describes the distribution of X as it hits the boundary of D.

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[edit] 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

\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, Hf(x) determines a non-negative Radon measure ω(x, D) on ∂D by

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 and the point x).

[edit] 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 an harmonic function of x ∈ D and
0 \leq \omega(x, D)(E) \leq 1;
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 ω(x, D)(E) is identically zero, in which case E is said to be a set of harmonic measure zero. Furthermore, if a compact subset K of Rn has harmonic measure zero with respect to some domain D, then it has harmonic measure zero with respect to any domain, and this situation arises if and only if K has zero harmonic capacity.

[edit] 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

\mu_{G}^{x} (F) = \mathbf{P}^{x} \big[ X_{\tau_{G}} \in F \big]

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

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