On the Almost Intersections of Transient Brownian Motions

Let {B1d(t)} and {Bd2(t)} be independent Brownian motions in Rd starting from 0 and nx respectively, and let wdi(a,b) ={x∈Rd: Bdi(t)=x for some t∈(a,b)}, i=1,2. Asymptotic expressions as n→∞ for the probability of dist(wd1(n2t1, n2t2), w2d(0,n2t3))≤1 with d≥4, respectively for the probability of dist(w14(n2t1,n2t2),w24(0,n2t3))≥1 are obtained. As an application, an improvement of a result due to M. Aizenman concerning the intersections of Wiener sausages in R4 is presented.