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.
For example, consider D = {[z.sub.0]} [subset] G and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the so-called harmonic measure of [z.sub.0].
where [w.sub.z] denotes the harmonic measure. Since D is compact, the latter implies that
This is the well-known characterization of the harmonic measure [rho] = [[omega].sub.z] by the mean-value principle.
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].sub.1] is not singular to harmonic measure on [[omega].sub.2].
Density at the boundary and harmonic measure. 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].
Harmonic measure; geometric and analytic points of view.