Chaos expansions of double intersection local time of Brownian motion in and renormalization

Double intersection local times [alpha](x,.) of Brownian motion which measure the size of the set of time pairs (s, t), s [not equal to] t, for which Wt and Ws + 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 eventually negative order Sobolev spaces with respect to the canonical Dirichlet structure on Wiener space. On the other hand, they offer an easy access to renormalization of [alpha](x,.) as x --> 0. The results, valid for any dimension d, describe a pattern in which the well known cases d = 2, 3 are naturally embedded.