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On a Stability Property of Harmonic Measures

  • Peter March
Chapter
Part of the Progress in Probability book series (PRPR, volume 17)

Abstract

The purpose of this note is to prove that the exit distributions of a diffusion from a somewhat smooth domain are stable under a large class of perturbations. That this need not be so in general had been observed already by M.V. Keldyš [5], who constructed a Jordan domain D containing the origin, with the property that if D n ,n ≥ 1, is any sequence of smooth domains such that
$$ \overline D \subset {D_n} \subset {D_{{n - 1}}}\;{\text{and}}\;\overline D = \mathop{ \cap }\limits_n {D_n} $$
(1)
then the classical harmonic measures h Dn (0,dy),n ≥ 1, do not converge weakly to the harmonic measure h D (0,dy). Of course, these measures are the exit distributions of Brownian motion starting from the origin.

Keywords

Brownian Motion Harmonic Measure Exit Time Smooth Domain Jordan Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  • Peter March
    • 1
  1. 1.Department of Mathematics & StatisticsCarleton UniversityOttawaCanada

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