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On the confinement property of two-dimensional Brownian motion among Poissonian obstacles. (English) Zbl 0753.60075
Let us distribute points in \({\mathfrak R}^ 2\) according to a Poisson point process with intensity \(\nu dx\), \(\nu>0\), and translate a given nonpolar compact subset \(K\) of \({\mathfrak R}^ 2\) to these points. Let \((Z,P)\) be a Brownian motion in \({\mathfrak R}^ 2\) killed when it hits any of these obstacles. Denote the life time of \(Z\) with \(T\). The problem is to get some information on how far from the origin the Brownian particle has visited up to time \(t\) under the condition that it is living at time \(t\). The exact main result is that \[ \lim_{t\to\infty}\mathbb{P}\oplus E_ 0(\sup_{0\leq s\leq t}| Z_ s|>2t^{1/4}(R_ 0+\exp(- \kappa(\log t)^{1/2})\mid T>t))=0, \] where \(\mathbb{P}\oplus E_ 0\) is the measure associated with the motion, \(\kappa\) is a suitable constant and \(R_ 0\) is a constant which can be computed explicitly. This work is intimately connected with [M. D. Donsker and S. R. S. Varadhan, Commun. Pure Appl. Math. 28, 525-565 (1975; Zbl 0333.60077)].
Reviewer: P.Salminen (Åbo)

60J65 Brownian motion
60H25 Random operators and equations (aspects of stochastic analysis)
Full Text: DOI
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