- Harmonic measure
In

mathematics ,**harmonic measure**is a concept that arises in the theory ofharmonic function s, where it can be used to estimate themodulus of ananalytic 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 anItō diffusion "X" describes the distribution of "X" as it hits the boundary of "D".**Definition**Let "D" be a bounded, open domain in "n"-

dimension alEuclidean space **R**^{"n"}, "n" ≥ 2, and let ∂"D" denote the boundary of "D". Anycontinuous function "f" : ∂"D" →**R**determines a uniqueharmonic function "H"_{"f"}that solves theDirichlet problem :$egin\{cases\}\; -\; Delta\; H\_\{f\}\; (x)\; =\; 0,\; x\; in\; D;\; \backslash \; H\_\{f\}\; (x)\; =\; f(x),\; x\; in\; partial\; D.\; end\{cases\}$

If a point "x" ∈ "D" is fixed, "H"

_{"f"}("x") determines a non-negativeRadon 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 arisesif and only if "K" has zeroharmonic 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, canonicalBrownian motion "B" on thereal 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:$mu\_\{G\}^\{x\}\; (F)\; =\; mathbf\{P\}^\{x\}\; ig\; [\; X\_\{\; au\_\{G\; in\; F\; ig]$

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 anopen ball centred on "x", then the harmonic measure of "B" on ∂"D" is invariant under allrotation s of "D" about "x" and coincides with the normalizedsurface 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.

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