On the almost intersections of transient Brownian motions

Let {B^d_1 (t )} and {B^d_2 (t )} be independent Brownian motions in R^d starting from 0 and nx respectively, and let w^d_i (a, b) = {x ∈ R^d : B^d_i (t ) = x for some t ∈ (a, b)}, i = 1, 2. Asymptotic expressions as n → ∞ for the probability of dist(w^d_1 (n^2 t_1, n^2 t_2), w^d_2 (0, n^2 t_3)) <= 1 with d >= 4, respectively for the probability of dist (w^4_1 (n^2 t_1, n^2 t_2),w^4_2 (0, n^2 t_3)) >= 1 are obtained. As an application, an improvement of a result due to M. Aizenman concerning the intersections of Wiener sausages in R^4 is presented.