harmonic measure


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harmonic measure

[här¦män·ik ′mezh·ər]
(mathematics)
Let D be a domain in the complex plane bounded by a finite number of Jordan curves Γ, and let Γ be the disjoint union of α and β, where α and β are Jordan arcs; the harmonic measure of α with respect to D is the harmonic function on D which assumes the value 1 on α and the value 0 on β.
References in periodicals archive ?
The topics include Brownian motion and harmonic measure in conic sections, the sharpness of certain approach regions, integral representation for space-time excessive functions, hyperbolic Riemann surfaces without unbounded positive harmonic functions, and a vanishing theorem on the point- wise defect of a rational iteration sequence for moving targets.
0]} [subset] G and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the so-called harmonic measure of [z.
This is the well-known characterization of the harmonic measure [rho] = [[omega].
The paper is organized into four sections: in the first the number of branches of [gamma] \ [gamma] is counted; in the second the theorem on proper maps is proved; in the third Stokes's theorem is proved; we end the paper with a discussion of density at the boundary and harmonic measure on [gamma].
The usefulness of this construction lies in the fact that harmonic measure on [[omega].
We obtain here a partial result about the two dimensional density at the boundary of [gamma]; we end the chapter with a brief discussion of the problem of harmonic measure on [gamma].
The hull of [gamma] has a nice property which is not known to be true in general: harmonic measure for a point in [gamma]\[gamma] is absolutely continuous with respect to [[lambda].
Given a rectifiable curve [gamma] whose hull is a Riemann surface which is regular for the Dirichlet problem, we would like to say that harmonic measure for a point p [element of ] [gamma]\[gamma] is absolutely continuous with respect to arclength on [gamma]; furthermore, it would be natural to hope that harmonic measure be given by integration of the normal derivative of the Green's function with respect to arclength on [gamma].