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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)

##### MSC:
 60J65 Brownian motion 60H25 Random operators and equations (aspects of stochastic analysis)
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##### References:
 [1] Bolthausen, Ann. Probab. 18 pp 1576– (1990) [2] , and , Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré, Probabilités et Statistiques, sup. au no. 2, 23, 1987, pp. 245–287. [3] Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984. [4] Donsker, Comm. Pure Appl. Math. 28 pp 525– (1975) [5] Dirichlet Forms and Markov Processes, North-Holland, Kodansha, Amsterdam, Tokyo, 1980. · Zbl 0422.31007 [6] Osserman, Amer. Math. Mon. 86 pp 1– (1979) [7] Schmock, Stochastics 29 pp 171– (1990) [8] Sznitman, J. Funct. Anal. 94 pp 223– (1990) [9] Sznitman, J. Funct. Anal. 94 pp 247– (1990) [10] Sznitman, Comm. Pure Appl. Math. 43 pp 809– (1990) [11] Potential Theory in Modern Function Theory, 2nd ed., Chelsea, New York, 1975. [12] Localization of a two-dimensional random walk with an attractive path interaction, Ann. Probab., to appear. · Zbl 0819.60028
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