View source for Harmonic measure

In [[mathematics]], especially [[potential theory]], '''harmonic measure''' is a concept related to the theory of [[harmonic function]]s that arises from the solution of the classical [[Dirichlet problem]]. [[File:BrownianMotionHarmonicMeasure.svg|thumb|Harmonic measure is the exit distribution of Brownian motion]] In [[probability theory]], the harmonic measure of a subset of the boundary of a bounded domain in [[Euclidean space]] <math>R^n</math>, <math>n\geq 2</math> 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 [[absolute value|modulus]] of an [[analytic function]] inside a domain ''D'' given bounds on the modulus on the [[boundary (topology)|boundary]] of the domain; a special case of this principle is [[Hadamard three-circle theorem|Hadamard's three-circle theorem]]. On simply connected planar domains, there is a close connection between harmonic measure and  the theory of [[conformal map]]s.

The term ''harmonic measure'' was introduced by [[Rolf Nevanlinna]] in 1928 for planar domains,<ref>R. Nevanlinna (1970), "Analytic Functions", Springer-Verlag, Berlin, Heidelberg, cf. Introduction p. 3</ref><ref>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.</ref> 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.<ref>{{cite journal
| last = Kakutani
| first = S.
| title = On Brownian motion in ''n''-space
| journal = Proc. Imp. Acad. Tokyo | volume = 20
| year = 1944
| pages = 648&ndash;652
 | doi=10.3792/pia/1195572742
| issue = 9
}}</ref>

==Definition==

Let ''D'' be a [[bounded set|bounded]], [[open set|open domain]] in ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>, ''n''&nbsp;&ge;&nbsp;2, and let &part;''D'' denote the boundary of ''D''.  Any [[continuous function]] ''f''&nbsp;:&nbsp;&part;''D''&nbsp;&rarr;&nbsp;'''R''' determines a unique [[harmonic function]] ''H''<sub>''f''</sub> that solves the [[Dirichlet problem]]

:<math>\begin{cases} - \Delta H_{f} (x) = 0, & x \in D; \\ H_{f} (x) = f(x), & x \in \partial D. \end{cases}</math>

If a point ''x''&nbsp;&isin;&nbsp;''D'' is fixed, by the [[Riesz–Markov–Kakutani representation theorem]] and the [[maximum principle]] ''H''<sub>''f''</sub>(''x'') determines a [[probability measure]] ''&omega;''(''x'',&nbsp;''D'') on &part;''D'' by

:<math>H_{f} (x) = \int_{\partial D} f(y) \, \mathrm{d} \omega(x, D) (y).</math>

The measure ''&omega;''(''x'',&nbsp;''D'') is called the '''harmonic measure''' (of the domain ''D'' with pole at ''x'').

==Properties==

* For any Borel subset ''E'' of &part;''D'', the harmonic measure ''&omega;''(''x'',&nbsp;''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''&nbsp;&sube;&nbsp;&part;''D'', ''&omega;''(''x'',&nbsp;''D'')(''E'') is an harmonic function of ''x''&nbsp;&isin;&nbsp;''D'' and

::<math>0 \leq \omega(x, D)(E) \leq 1;</math>
::<math>1 - \omega(x, D)(E) = \omega(x, D)(\partial D \setminus E);</math>

:Hence, for each ''x'' and ''D'', ''&omega;''(''x'',&nbsp;''D'') is a [[probability measure]] on &part;''D''.

* If ''&omega;''(''x'',&nbsp;''D'')(''E'')&nbsp;=&nbsp;0 at even a single point ''x'' of ''D'', then <math>y \mapsto\omega(y,D)(E)</math> 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''':<ref>F. and M. Riesz (1916), "Über die Randwerte einer analytischen Funktion", Quatrième Congrès des Mathématiciens Scandinaves, Stockholm, pp. 27–44.</ref> If <math>D\subset\mathbb{R}^2</math> is a simply connected planar domain bounded by a [[Arc length|rectifiable curve]] (i.e. if <math>H^1(\partial D)<\infty</math>), then harmonic measure is mutually absolutely continuous with respect to arc length: for all <math>E\subset\partial D</math>, <math>\omega(X,D)(E)=0</math> if and only if <math>H^1(E)=0</math>.
* '''Makarov's theorem''':<ref>{{cite journal
| last = Makarov
| first = N. G.
| title = On the Distortion of Boundary Sets Under Conformal Maps
| journal = Proc. London Math. Soc. | series = 3 | volume = 52
| issue = 2
| year = 1985
| pages = 369&ndash;384
| doi = 10.1112/plms/s3-51.2.369
}}</ref> Let <math>D\subset\mathbb{R}^2</math> be a simply connected planar domain. If <math>E\subset\partial D</math> and <math>H^s(E)=0</math> for some <math>s<1</math>, then <math>\omega(x,D)(E)=0</math>. Moreover, harmonic measure on ''D'' is [[Singular measure|mutually singular]] with respect to ''t''-dimensional Hausdorff measure for all&nbsp;''t''&nbsp;>&nbsp;1.

* '''Dahlberg's theorem''':<ref>{{cite journal
| last = Dahlberg
| first = Björn E. J.
| title = Estimates of harmonic measure
| journal = Arch. Rat. Mech. Anal. | volume = 65
| issue = 3
| year = 1977
| pages = 275&ndash;288
| doi = 10.1007/BF00280445
 | bibcode=1977ArRMA..65..275D
}}</ref> If <math>D\subset\mathbb{R}^n</math> is a bounded [[Lipschitz domain]], then harmonic measure and (''n''&nbsp;&minus;&nbsp;1)-dimensional Hausdorff measure are mutually absolutely continuous: for all <math>E\subset\partial D</math>, <math>\omega(X,D)(E)=0</math> if and only if <math>H^{n-1}(E)=0</math>.

==Examples==

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

==The harmonic measure of a diffusion==

Consider an '''R'''<sup>''n''</sup>-valued Itō diffusion ''X'' starting at some point ''x'' in the interior of a domain ''D'', with law '''P'''<sup>''x''</sup>.  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 (mathematics)|interval]] (&minus;1,&nbsp;+1) at &minus;1 with probability ½ and at +1 with probability ½, so ''B''<sub>''&tau;''<sub>(&minus;1,&nbsp;+1)</sub></sub> is [[uniform distribution (discrete)|uniformly distributed]] on the set {&minus;1,&nbsp;+1}.

In general, if ''G'' is [[compactly embedded]] within '''R'''<sup>''n''</sup>, then the '''harmonic measure''' (or '''hitting distribution''') of ''X'' on the boundary &part;''G'' of ''G'' is the measure ''&mu;''<sub>''G''</sub><sup>''x''</sup> defined by

:<math>\mu_{G}^{x} (F) = \mathbf{P}^{x} \big[ X_{\tau_{G}} \in F \big]</math>

for ''x''&nbsp;&isin;&nbsp;''G'' and ''F''&nbsp;&sube;&nbsp;&part;''G''.

Returning to the earlier example of Brownian motion, one can show that if ''B'' is a Brownian motion in '''R'''<sup>''n''</sup> starting at ''x''&nbsp;&isin;&nbsp;'''R'''<sup>''n''</sup> and ''D''&nbsp;&sub;&nbsp;'''R'''<sup>''n''</sup> is an [[open ball]] centred on ''x'', then the harmonic measure of ''B'' on &part;''D'' is [[invariant measure|invariant]] under all [[rotation]]s of ''D'' about ''x'' and coincides with the normalized [[surface measure]] on &part;''D''

==General references==

* {{cite book
| last1 = Garnett
| first1 = John B.
| last2 = Marshall
| first2 = Donald E.
| title = Harmonic Measure
| publisher = Cambridge University Press
| location = Cambridge
| year = 2005
| isbn = 978-0-521-47018-6
| doi = 10.2277/0521470188
}}

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

* {{cite book
| last1 = Capogna
| first1 = Luca 
| last2 = Kenig
| first2 = Carlos E.
| last3 = Lanzani
| first3 = Loredana
| title = Harmonic  Measure: Geometric and Analytic Points of View
| publisher= American Mathematical Society
| series = University Lecture Series
| volume = ULECT/35
| pages = 155
| year = 2005
| isbn = 978-0-8218-2728-4
}}

==References==

<references/>
* 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.PreissandT. Toro, Boundary Structure and Size in terms of Interior and Exterior Harmonic Measures in Higher Dimensions, Jour. ofAmer. Math. Soc.vol22  July 2009,  no3,771-796
* S .G.Krantz,  The Theory and Practice of Conformal Geometry,  Dover Publ.Mineola New York (2016) esp. Ch6 classical case

==External links==
* {{springer
 | title = Harmonic measure
 | id = H/h046500
 | last = Solomentsev
 | first = E.D.
}}

[[Category:Measures (measure theory)]]
[[Category:Potential theory]]