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

Double intersection local times α(x,.) of Brownian motion W if Rd which measure the size of the set of time pairs (s, t), s ≠ 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 α(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 α(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.