Abstract.
Consider Brownian motion among random obstacles obtained by translating a fixed compact nonpolar subset of ℝd, d≥ 1, at the points of a Poisson cloud of constant intensity v <: 0. Assume that Brownian motion is absorbed instantaneously upon entering the obstacle set. In SZN-conf Sznitman has shown that in d = 2, conditionally on the event that the process does not enter the obstacle set up to time t, the probability that Brownian motion remains within distance ∼t 1/4 from its starting point is going to 1 as t goes to infinity. We show that the same result holds true for d≥ 3, with t 1/4 replaced by t 1/( d +2). The proof is based on Sznitmans refined method of enlargement of obstacles [10] as well as on a quantitative isoperimetric inequality due to Hall [4].
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Received: 6 July 1998
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Povel, T. Confinement of Brownian motion among Poissonian obstacles in ℝd, d≥ 3. Probab Theory Relat Fields 114, 177–205 (1999). https://doi.org/10.1007/s440-1999-8036-0
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DOI: https://doi.org/10.1007/s440-1999-8036-0
- Mathematics Subject Classification (1991): 60K40, 60G17, 82D30