Critical dimensions for the existence of self-intersection local times of the N-parameter Brownian motion in R-d

Fix two rectangles A, B in [0, 1](N). Then the size of the random set of do uble points of the N-parameter Brownian motion (W-t)(t is an element of[0, 1])(N) in R-d, i.e, the ser of pairs (s, t), where s is an element of A, t is an element of B, and W-s = W-t, can be measured as usual by a self-inter section local time. IF A = B, we show that the critical dimension below whi ch self-intersection local time does not explode, is given by d = 2N. If A boolean AND B is a p-dimensional rectangle, it is 4N-2p (0 less than or equ al to p less than or equal to N). If A boolean AND B = O, it is infinite. I n all cases, we derive the rate of explosion of canonical approximations of self-intersection local time for dimensions above the critical one; and de termine its smoothness in terms of the canonical Dirichlet structure on Wie ner space.