Self-intersection local times, occupation fields, and stochastic integrals

Abstract

where functions 6,; converge to 6 as &JO. It turns out that B(D) = co for every open set D which intersects with the diagonal (t, = t2 }. However, Varadhan [V] has shown that B,:( D,) u log(s ’ ) converges to a finite limit as E JO if D, = ‘0 <s < t < u}. Rosen [R2] has found an expression for the limit in terms 1 of Ito’s stochastic integrals. The integrals (0.1) and their renormalizations have been studied also in [Y 1, L 1, L2] and, for the Brownian motion in lR3, in [ Rl, R3, W, Y23. 0.2. In this paper we investigate more general self-intersection local times