On the means of approach to the boundary of Brownian motion. (English) Zbl 0628.60076

Cranston, M.

Ann. Probab. 15, 1009-1013 (1987).

This paper studies the behavior of a planar Brownian motion as it exists from a simply connected Greenian domain D. The technique is to fix a point \(\xi\) in the Martin boundary of D, and take u to be the Martin kernel with pole at \(\xi\), v its harmonic conjugate, X the u-processes (Brownian motion conditioned to exit D at \(\xi)\), and then to investigate the behavior of the process \((u(X_ t),v(X_ t))\) at the exit time from D. Specializing to the domain \(D=\{z:| z| <1\}\), the results of this paper are used to discuss the connection between the classical and probabilistic Fatou theorems.

Reviewer: J.Mitro

MSC:

60J45	Probabilistic potential theory
60J65	Brownian motion

Keywords:

planar Brownian motion; Martin boundary; Martin kernel; probabilistic Fatou theorems