Non-intersection exponents for Brownian paths. Part II: Estimations and applications to a random fractal.

Abstract

Let X and Y be independent two-dimensional Brownian motions, X(0) = (0; 0); Y(0) = ([epsilon]; 0), and let p([epsilon]) = P(X[0; 1] [intersected with] Y [0; 1] = [empty set], q([epsilon]) = {Y [0; 1] does not contain a closed loop around 0}. Asymptotic estimates (when [epsilon] --> 0) of p([epsilon]); q([epsilon]), and some related probabilities, are given. Let F be the boundary of the unbounded connected component of [the set of real numbers squared]\Z[0; 1], where Z(t) = X(t) - tX(1) for t [is an element of the set] [0; 1]. Then F is a closed Jordan arc and the Hausdorff dimension of F is less or equal to 3/2 - 1=(4[pi squared]).