×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.