Critical Dimensions for the Existence of Self-intersection Local times of the N-parameter Brownian Motion in R

Abstract. Fix two rectangles A, B in [0, 1] . Then the size of the random set of double points of the N -parameter Brownian motion (Wt)t∈[0,1]N in R, 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.