Critical Dimensions for the Existence of Self-Intersection Local Times of the N-Parameter Brownian Motion in Rd

Abstract

Fix two rectangles A, B in [0, 1]N. Then the size of the random set of double points of the N-parameter Brownian motion \((W_t )_{t \in } [0,1]^N \) in Rd, i.e, the set of pairs (s, t), where s∈A, t∈B, and Ws=Wt, can be measured as usual by a self-intersection local time. If A=B, we show that the critical dimension below which self-intersection local time does not explode, is given by d=2N. If A ∩ B is a p-dimensional rectangle, it is 4N–2p (0≤p≤N). If A ∩ B = ∅, it is infinite. In all cases, we derive the rate of explosion of canonical approximations of self-intersection local time for dimensions above the critical one, and determine its smoothness in terms of the canonical Dirichlet structure on Wiener space.