Harmonic measure

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".

Definition

Let "D" be a bounded, open domain in "n"-dimensional Euclidean space R^"n", "n" ≥ 2, and let ∂"D" denote the boundary of "D". Any continuous function "f" : ∂"D" → R determines a unique harmonic function "H"_"f" that solves the Dirichlet problem

:

If a point "x" ∈ "D" is fixed, "H"_"f"("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").

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 R^"n" 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.

The harmonic measure of a diffusion

Consider an R^"n"-valued Itō diffusion "X" starting at some point "x" in the interior of a domain "D", with law P^"x". 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 "B"_{"τ"(−1, +1)} is uniformly distributed on the set {−1, +1}.

In general, if "G" is compactly embedded within R^"n", then the harmonic measure (or hitting distribution) of "X" on the boundary ∂"G" of "G" is the measure "μ"_"G"^"x" 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 R^"n" starting at "x" ∈ R^"n" and "D" ⊂ R^"n" 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"

References

* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
publisher=Springer
location = Berlin
year = 2003
id = ISBN 3-540-04758-1 MathSciNet|id=2001996 (See Sections 7, 8 and 9)

External links

* springer
title = Harmonic measure
id = H/h046500
last = Solomentsev
first = E.D.