Wiener sausage and self-intersection local times

Abstract

Let B = ( B t , t ⩾ 0) be a standard Brownian motion in R 2. For every ε > 0 and every compact subset K of R 2, the Wiener sausage of radius ε associated with K is defined as the union of the sets B s + εK, s ϵ [0, 1]. The present paper gives full asymptotic expansions for the area of the Wiener sausage, when the radius ε goes to 0. The kth term of the expansion is of order ¦log ε¦ −k and involves a random variable which measures the number of k-multiple self-intersections of the process. Such random variables are called (renormalized) self-intersection local times and have been recently introduced and studied by E. B. Dynkin. A self-contained construction of these local times is given, together with a number of new approximations.