# Harmonic measure

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In [mathematics](/source/mathematics), especially [potential theory](/source/potential_theory), '''harmonic measure''' is a concept related to the theory of [harmonic function](/source/harmonic_function)s that arises from the solution of the classical [Dirichlet problem](/source/Dirichlet_problem). 300px|thumb|Harmonic measure is the exit distribution of Brownian motion In [probability theory](/source/probability_theory), the harmonic measure of a subset of the boundary of a bounded domain in [Euclidean space](/source/Euclidean_space) <math>R^n</math>, <math>n\geq 2</math> is the probability that a [Brownian motion](/source/Brownian_motion) started inside a domain hits that subset of the boundary. More generally, harmonic measure of an [Itō diffusion](/source/It%C5%8D_diffusion) ''X'' describes the distribution of ''X'' as it hits the boundary of ''D''. In the [complex plane](/source/complex_plane), harmonic measure can be used to estimate the [modulus](/source/absolute_value) of an [analytic function](/source/analytic_function) inside a domain ''D'' given bounds on the modulus on the [boundary](/source/boundary_(topology)) of the domain; a special case of this principle is [Hadamard's three-circle theorem](/source/Hadamard_three-circle_theorem). On simply connected planar domains, there is a close connection between harmonic measure and  the theory of [conformal map](/source/conformal_map)s.

The term ''harmonic measure'' was introduced by [Rolf Nevanlinna](/source/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, [Frigyes Riesz](/source/Frigyes_Riesz), [Marcel Riesz](/source/Marcel_Riesz), [Torsten Carleman](/source/Torsten_Carleman), [Alexander Ostrowski](/source/Alexander_Ostrowski) and [Gaston Julia](/source/Gaston_Julia). The connection between harmonic measure and Brownian motion was first identified by Kakutani 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
| doi-access = free
}}</ref>

==Definition==

Let ''D'' be a [bounded](/source/bounded_set), [open domain](/source/open_set) in ''n''-[dimension](/source/dimension)al [Euclidean space](/source/Euclidean_space) '''R'''<sup>''n''</sup>, ''n''&nbsp;&ge;&nbsp;2, and let &part;''D'' denote the boundary of ''D''.  Any [continuous function](/source/continuous_function) ''f''&nbsp;:&nbsp;&part;''D''&nbsp;→&nbsp;'''R''' determines a unique [harmonic function](/source/harmonic_function) ''H''<sub>''f''</sub> that solves the [Dirichlet problem](/source/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](/source/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem) and the [maximum principle](/source/maximum_principle) ''H''<sub>''f''</sub>(''x'') determines a [probability measure](/source/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](/source/indicator_function) of ''E''.
* For fixed ''D'' and ''E''&nbsp;&sube;&nbsp;&part;''D'', ''&omega;''(''x'',&nbsp;''D'')(''E'') is a 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](/source/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](/source/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 [rectifiable curve](/source/Arc_length) (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 [mutually singular](/source/Singular_measure) 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
| s2cid = 120614580
}}</ref> If <math>D\subset\mathbb{R}^n</math> is a bounded [Lipschitz domain](/source/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 derivative](/source/Radon%E2%80%93Nikodym_theorem) <math>d\omega(X,\mathbb{D})/dH^1</math> is called the [Poisson kernel](/source/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](/source/Hausdorff_measure)) on the unit sphere <math>S^{n-1}</math> and <math>H^{n-1}(S^{n-1})=\sigma_{n-1}</math>.
* 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](/source/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 map](/source/Riemann_mapping_theorem) which sends the origin to ''X'', i.e. <math>f(0)=X</math>. See [Carathéodory's theorem](/source/Carath%C3%A9odory's_theorem_(conformal_mapping)).
* If <math>D\subset\mathbb{R}^2</math> is the domain bounded by the [Koch snowflake](/source/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](/source/real_line) starting at 0 exits the [interval](/source/interval_(mathematics)) (&minus;1,&nbsp;+1) at &minus;1 with probability {{sfrac|1|2}} and at +1 with probability {{sfrac|1|2}}, so ''B''<sub>''&tau;''<sub>(&minus;1,&nbsp;+1)</sub></sub> is [uniformly distributed](/source/uniform_distribution_(discrete)) on the set {&minus;1,&nbsp;+1}.

In general, if ''G'' is [compactly embedded](/source/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](/source/open_ball) centred on ''x'', then the harmonic measure of ''B'' on &part;''D'' is [invariant](/source/invariant_measure) under all [rotation](/source/rotation)s of ''D'' about ''x'' and coincides with the normalized [surface measure](/source/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
}}

* {{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 | author3-link = Loredana Lanzani
| 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. Preissand, T. Toro, Boundary Structure and Size in terms of Interior and Exterior Harmonic Measures in Higher Dimensions, Jour. of Amer. Math. Soc. vol 22  July 2009,  no3,771-796
* S. G. Krantz, The Theory and Practice of Conformal Geometry,  Dover Publ. Mineola New York (2016) esp. Ch 6 classical case

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

{{Measure theory}}

Category:Measures (measure theory)
Category:Potential theory

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