CHAOS EXPANSIONS OF DOUBLE INTERSECTION LOCAL TIME OF BROWNIAN-MOTIONIN R(D) AND RENORMALIZATION

Double intersection local times alpha(x,.) of Brownian motion W in R(d ) which measure the size of the set of time pairs (s, t), s not equal t, for which W-t and W-s + x coincide can be developed into series of multiple Wiener-Ito integrals. These series representations reveal on the one hand the degree of smoothness of alpha(x,.) in terms of eventu ally negative order Sobolev spaces with respect to the canonical Diric hlet structure on Wiener space. On the other hand, they offer an easy access to renormalization of alpha(x,.) as \x\ --> 0. The results, val id for any dimension d, describe a pattern in which the well known cas es d = 2, 3 are naturally embedded.