Vol. 95, No. 1, 1981

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ISSN: 0030-8730
Brownian motion and sets of harmonic measure zero

Bernt Karsten Oksendal

Vol. 95 (1981), No. 1, 179–192
Abstract

Using Brownian motion the following results are established:

  1. Harmonic measure and Keldysh measure are always singular with respect to area measure in the plane. More generally, this holds for the distribution of the first exit point for Brownian motion of a given Borel set.
  2. If U is open and K ∂U is compact, then K has harmonic measure 0 w.r.t. U if ∂U satisfies a certain metric density condition at each point of K and, in addition, K satisfies one of the following two conditions:
    1. K has zero length and is lying on a straight line or
    2. K has α-dimensional Hausdorff measure zero, for some α < 12.

Mathematical Subject Classification 2000
Primary: 60J45
Secondary: 31A15
Milestones
Received: 21 February 1980
Revised: 25 August 1980
Published: 1 July 1981
Authors
Bernt Karsten Oksendal