Non-intersection exponents for Brownian paths. Part I: Existence and an invariance principle

Abstract

Let X and Y be independent three-dimensional Brownian motions, X(0) = (0; 0; 0), Y (0) = (1; 0; 0) and let p [subscript]r = P(X[0; r] [intersected with] Y [0; r] = [empty set]. Then the "non- intersection exponent" [from] lim [subscript]r [to infinity] -log p [subscript]r / log r exists and is equal to a similar "non-intersection exponent" for random walks. Analogous results hold in [the set of real numbers squared] and for more than two paths.