Harmonic measure

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, harmonic measure of a subset of the boundary of a bounded domain in Euclidean space $R^{n}$ , $n\geq 2$ 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,^[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]

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 H_f 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, by the Riesz representation theorem and the maximum principle H_f(x) determines a probability 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 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 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 $y\mapsto \omega (y,D)(E)$ 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.

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

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

Examples

If ${\mathbb {D}}=\{X\in {\mathbb {R}}^{2}:|X|<1\}$ is the unit disk, then harmonic measure of $\mathbb {D}$ with pole at the origin is length measure on the unit circle normalized to be a probability, i.e. $\omega (0,{\mathbb {D}})(E)=|E|/2\pi$ for all $E\subset S^{1}$ where $|E|$ denotes the length of $E$ .
If $\mathbb {D}$ is the unit disk and $X\in {\mathbb {D}}$ , then $\omega (X,{\mathbb {D}})(E)=\int _{E}{\frac {1-|X|^{2}}{|X-Q|^{2}}}{\frac {dH^{1}(Q)}{2\pi }}$ for all $E\subset S^{1}$ where $H^{1}$ denotes length measure on the unit circle. The Radon–Nikodym derivative $d\omega (X,{\mathbb {D}})/dH^{1}$ is called the Poisson kernel.
More generally, if $n\geq 2$ and ${\mathbb {B}}^{n}=\{X\in {\mathbb {R}}^{n}:|X|<1\}$ is the n-dimensional unit ball, then harmonic measure with pole at $X\in {\mathbb {B}}^{n}$ is $\omega (X,{\mathbb {B}}^{n})(E)=\int _{E}{\frac {1-|X|^{2}}{|X-Q|^{n}}}{\frac {dH^{{n-1}}(Q)}{\sigma _{{n-1}}}}$ for all $E\subset S^{{n-1}}$ where $H^{{n-1}}$ denotes surface measure ((n − 1)-dimensional Hausdorff measure) on the unit sphere $S^{{n-1}}$ and $H^{{n-1}}(S^{{n-1}})=\sigma _{{n-1}}$ .
Harmonic Measure on Simply Connected Planar Domains
If $D\subset {\mathbb {R}}^{2}$ is a simply connected planar domain bounded by a Jordan curve and X $\in$ D, then $\omega (X,D)(E)=|f^{-1}(E)|/2\pi$ for all $E\subset \partial D$ where $f:\mathbb {D} \rightarrow D$ is the unique Riemann map which sends the origin to X, i.e. $f(0)=X$ . See Carathéodory's theorem.
If $D\subset {\mathbb {R}}^{2}$ is the domain bounded by the Koch snowflake, then there exists a subset $E\subset \partial D$ of the Koch snowflake such that $E$ has zero length ( $H^{1}(E)=0$ ) and full harmonic measure $\omega (X,D)(E)=1$ .

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 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 Rⁿ, then the harmonic measure (or hitting distribution) of X on the boundary ∂G of G is the measure μ_G^x 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

General references

Garnett, John B.; Marshall, Donald E. (2005). Harmonic Measure. Cambridge: Cambridge University Press. doi:10.2277/0521470188. 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)

References

↑ R. Nevanlinna (1970), "Analytic Functions", Springer-Verlag, Berlin, Heidelberg, cf. Introduction p. 3
↑ 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.
↑ Kakutani, S. (1944). "On Brownian motion in n-space". Proc. Imp. Acad. Tokyo. 20 (9): 648–652. doi:10.3792/pia/1195572742.
↑ F. and M. Riesz (1916), "Über die Randwerte einer analytischen Funktion", Quatrième Congrès des Mathématiciens Scandinaves, Stockholm, pp. 27–44.
↑ 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.
↑ 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.

External links

Solomentsev, E.D. (2001), "Harmonic measure", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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