Confinement of Brownian motion among Poissonian obstacles in ℝd, d≥ 3

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 ∼t1/4 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 t1/4 replaced by t1/(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].