Memoirs of the American Mathematical Society 1999; 125 pp; softcover Volume: 142 ISBN10: 0821813404 ISBN13: 9780821813409 List Price: US$48 Individual Members: US$28.80 Institutional Members: US$38.40 Order Code: MEMO/142/675
 Sufficient conditions are obtained for the continuity of renormalized selfintersection local times for the multiple intersections of a large class of strongly symmetric Lévy processes in \(R^m\), \(m=1,2\). In \(R^2\) these include Brownian motion and stable processes of index greater than 3/2, as well as many processes in their domains of attraction. In \(R^1\) these include stable processes of index \(3/4<\beta\le 1\) and many processes in their domains of attraction. Let \((\Omega,\mathcal F(t),X(t), P^{x})\) be one of these radially symmetric Lévy processes with 1potential density \(u^1(x,y)\). Let \(\mathcal G^{2n}\) denote the class of positive finite measures \(\mu\) on \(R^m\) for which \(\int\!\!\int (u^1(x,y))^{2n}\,d\mu(x)\,d\mu(y)<\infty.\) For \(\mu\in\mathcal G^{2n}\), let \[\alpha_{n,\epsilon}(\mu,\lambda) \overset{\text{def}}{=}\int\!\!\int_{\{0\leq t_1\leq \cdots \leq t_n\leq \lambda\}} f_{\epsilon}(X(t_1)x)\prod_{j=2}^n f_{\epsilon}(X(t_j) X(t_{j1}))\,dt_1\cdots\,dt_n \,d\mu(x)\] 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 selfintersection local time of \(X\) with respect to the measure \(\mu\) is defined as \[\gamma_{n}(\mu)=\lim_{\epsilon\to 0}\,\sum_{k=0}^{n1}(1)^{k} {n1 \choose k}(u^1_{\epsilon}(0))^{k} \alpha_{nk,\epsilon}(\mu,\lambda)\] where \(u^1_{\epsilon}(x)\overset{\text{def}}{=} \int f_{\epsilon}(xy)u^1(y)\,dy\), with \(u^1(x)\overset{\text{def}}{=} u^1(x+z,z)\) for all \(z\in R^m\). Conditions are obtained under which this limit exists in \(L^2(\Omega\times R^+,P^y_\lambda)\) for all \(y\in R^m\), where \(P^y_\lambda\overset{\text{def}}{=} P^y\times P_\lambda\). Let \(\{\mu_x,x\in R^m\}\) denote the set of translates of the measure \(\mu\). The main result in this paper is a sufficient condition for the continuity of \(\{\gamma_{n}(\mu_x),\,x\in R^m\}\) namely that this process is continuous \(P^y_\lambda\) almost surely for all \(y\in R^m\), if the corresponding 2\(n\)th Wick power chaos process, \(\{:G^{2n}\mu_x:,\,x\in R^m\}\) is continuous almost surely. This chaos process is obtained in the following way. A Gaussian process \(G_{x,\delta}\) is defined which has covariance \(u^1_\delta(x,y)\), where \(\lim_{\delta\to 0}u_\delta^1(x,y)=u^1(x,y)\). Then \(:G^{2n}\mu_x:\overset{\text{def}}{=} \lim_{\delta\to 0}\int :G_{y,\delta}^{2n}:\,d\mu_x(y)\) where the limit is taken in \(L^2\). (\(:G_{y,\delta}^{2n}:\) is the 2\(n\)th Wick power of \(G_{y,\delta}\), that is, a normalized Hermite polynomial of degree 2\(n\) in \(G_{y,\delta}\).) This process has a natural metric \[\begin{aligned} d(x,y)&\overset{\text{def}}{=} \frac1{(2n)!}\left(E(:G^{2n}\mu_x::G^{2n}\mu_y:)^2\right)^{1/2}\\ & =\left(\int\!\! \int \left(u^1(u,v)\right)^{2n} \left( d(\mu_x(u)\mu_y(u)) \right) \left(d(\mu_x(v)\mu_y(v)) \right)\right)^{1/2}\,. \end{aligned}\] A well known metric entropy condition with respect to \(d\) gives a sufficient condition for the continuity of \(\{:G^{2n}\mu_x:,\,x\in R^m\}\) and hence for \(\{\gamma_{n}(\mu_x),\,x\in R^m\}\). Readership Graduate students and research mathematicians interested in probability. Table of Contents  Introduction
 Wick products
 Wick power chaos processes
 Isomorphism theorems
 Equivalence of two versions of renormalized selfintersection local times
 Continuity
 Stable mixtures
 Examples
 A large deviation result
 Appendix A. Necessary conditions
 Appendix B. The case \(n=3\)
 Bibliography
