Brownian paths and cones. (English) Zbl 0574.60053

Burdzy, Krzysztof

Ann. Probab. 13, 1006-1010 (1985).

Let \({\mathcal W}(\alpha)\) be the totality of cones of vertical angle \(\alpha\) in \({\mathbb{R}}^ n\), and Z(t) be the standard n-dimensional Brownian motion. Then the author proves
Theorem 1. Let \(A(\alpha)\equiv \{\omega\); there exist \(s_ 1\) and \(s_ 2\), \(0\leq s_ 1\leq s_ 2\) and \(w={\mathcal W}(\alpha)\) such that \(v(W)=Z(s_ 1)\) and Z(t)\(\in W\) for all \(t\in (s_ 1,s_ 2)\}\), where v(W) means the vertex point of the cone W. Then \(P(A(\alpha))=0\), if \(\cos (\alpha)>1/\sqrt{n}\), and \(=1\), if \(\cos (\alpha)<1/\sqrt{n}.\)
He also claims without proof that the following fact holds
Theorem 2. Fix a (n-1)-dimensional hyperplane H in \({\mathbb{R}}^ n\) and set \(B(\alpha)=\{\omega\); there exist \(s_ 1\), \(s_ 2\), \(0<s_ 1<s_ 2\), and \(W\in {\mathcal W}(\alpha)\) such that \(W\cap H=\emptyset\), \(v(W)=Z(s_ 1)\in H\) and Z(t)\(\in W\) for all \(t\in (s_ 1,s_ 2)\}\). Then \(P(B(\alpha))=0\) for any \(\alpha\), \(0<\alpha <\pi /2\).

Reviewer: S.Takenaka

MSC:

60G17	Sample path properties
60J65	Brownian motion
60G40	Stopping times; optimal stopping problems; gambling theory

Keywords:

Brownian exit time; Brownian path