Renormalized self-intersection local times and Wick power chaos processes

Sufficient conditions are obtained for the continuity of renormalized self- intersection local times for the multiple intersections of a large class of strongly symmetric Levy processes in R-m, m = 1, 2. In R-2 these include B rownian motion and stable processes of index greater than 3/2, as well as m any processes in their domains of attraction. In R-1 these include stable p rocesses of index 3/4 < beta less than or equal to 1 and many processes in their domains of attraction. Let (Omega, F(t), X(t), P-x) be one of these radially symmetric Levy proces ses with 1-potential density u(1)(x, y). Let G(2n) denote the class of posi tive finite measures mu on R-m for which integral integral (u(1) (x, y))(2n) d mu(x) d mu(y) < infinity. For mu is an element of G(2n), let alpha(n,epsilon)(mu, lambda) <(def)double under bar> integral integral({0 l ess than or equal to t1 less than or equal to...less than or equal to tn le ss than or equal to lambda}) [GRAPHICS] where f(epsilon) is an approximate delta-function at zero and lambda is an random exponential time, with mean one, independent of X, with probability measure P-lambda. The renormalized self-intersection local time of X with r espect to the measure mu is defined as [GRAPHICS] where u(epsilon)(1)(x) <(def)double under bar> integral f(epsilon)(x - y)u( 1)(y) dy, with u(1)(x) <(def)double under bar> u(1)(x + z, z) for all z is an element of R-m. Conditions are obtained under which this limit exists in L-2(Omega x R+, P-lambda(y)) for all y is an element of R-m, where P-lambd a(y) <(def)double under bar> P-y x P-lambda. Let {mu(x), x is an element of R-m} denote the set of translates of the mea sure mu. The main result in this paper is a sufficient condition for the co ntinuity of {gamma(n)(mu(x)), x is an element of R-m} namely that this process is continuous P-lambda(y) almost surely for all y is an element of R-m, if the corresponding 2n-th Wick power chaos process, {: G(2n)mu(x) :, x is an element of R-m} is continuous almost surely. This chaos process is obtained in the following way. A Gaussian process G(x,delt a) is defined which has covariance u(delta)(1)(x, y), where lim(delta-->0) u(delta)(1)(x, y) = u(1)(x, y). Then [GRAPHICS] where the limit is taken in L-2. (: G(y,delta)(2n) : is the 2n-th Wick powe r of G(y,delta), that is, a normalized Hermite polynomial of degree 2n in G (y,delta)). This process has a natural metric d(x, y) <(def)double under bar> 1/(2n)! (E(: G(2n)mu(x) : - : G(2n)mu(y) :) (2))(1/2) =(integral integral (u(1)(u, v))(2n) (d(mu(x)(u) - mu(y)(u))) (d(mu(x)(v) - mu(y)(v))))(1/2). A well known metric entropy condition with respect to d gives a sufficient condition for the continuity of {: G(2n)mu(x) :, x is an element of R-m} an d hence for {gamma(n)(mu(x)), x is an element of R-m}.