About: Harmonic measure

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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. In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space , 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, the

Attributes	Values
rdfs:label	Harmonic measure (en) 调和测度 (zh)
rdfs:comment	數學中，調和測度是調和函數理論中出現的一個概念。给定了一个解析函数的模在一个区域 D 边界上的界，能用调和测度去估计函数在区域内部的模。在一个非常相关的领域，一个 X 的调和测度描绘了 X 撞击 D 边界的分布。 (zh) 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. In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space , 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, the (en)
foaf:depiction
dcterms:subject	Potential theory Measures (measure theory)
Wikipage page ID	12141074 (xsd:integer)
Wikipage revision ID	1091209997 (xsd:integer)
Link from a Wikipage to another Wikipage	Rolf Nevanlinna Arc length Riesz–Markov–Kakutani representation theorem Invariant measure Complex plane Continuous function Analytic function Potential theory Mathematics Maximum principle Boundary (topology) Bounded set Conformal map Harnack's inequality Hadamard three-circle theorem Hausdorff measure Lipschitz domain Absolute value Euclidean space Carathéodory's theorem (conformal mapping) Dirichlet problem Koch snowflake Itō diffusion Probability theory Harmonic function Interval (mathematics) Measures (measure theory) Potential theory Jordan curve Dimension Indicator function Brownian motion Open set Radon–Nikodym theorem Rotation Singular measure Uniform distribution (discrete) Compactly embedded Poisson kernel Probability measure Real line Riemann mapping theorem Open ball dbr:Surface_measure
sameAs	Harmonic measure Harmonic measure Harmonic measure Harmonic measure
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Cite_book dbt:MathSciNet dbt:Measure_theory
thumbnail	wiki-commons:Special:FilePath/BrownianMotionHarmonicMeasure.svg?width=300
first	E.D. (en)
id	H/h046500 (en)
last	Solomentsev (en)
title	Harmonic measure (en)
has abstract	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. In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space , 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, 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. (en) 數學中，調和測度是調和函數理論中出現的一個概念。给定了一个解析函数的模在一个区域 D 边界上的界，能用调和测度去估计函数在区域内部的模。在一个非常相关的领域，一个 X 的调和测度描绘了 X 撞击 D 边界的分布。 (zh)
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page length (characters) of wiki page	11519 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Harmonic_measure
is Link from a Wikipage to another Wikipage of	Rolf Nevanlinna Muckenhoupt weights Walk-on-spheres method Balayage Nikolai Georgievich Makarov Isodynamic point Random walk Planar Riemann surface
is foaf:primaryTopic of	wikipedia-en:Harmonic_measure

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