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Renormalized Self-Intersection Local Times and Wick Power Chaos Processes
Michael B. Marcus, City College of New York, NY, and Jay Rosen, College of Staten Island, NY

Memoirs of the American Mathematical Society
1999; 125 pp; softcover
Volume: 142
ISBN-10: 0-8218-1340-4
ISBN-13: 978-0-8218-1340-9
List Price: US$48
Individual Members: US$28.80
Institutional Members: US$38.40
Order Code: MEMO/142/675
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Sufficient conditions are obtained for the continuity of renormalized self-intersection 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 1-potential 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 \[\alpha_{n,\epsilon}(\mu,\lambda) \overset{\text{def}}{=}\int\!\!\int_{\{0\leq t_1\leq \cdots \leq t_n\leq \lambda\}} f_{\epsilon}(X(t_1)-x)\prod_{j=2}^n f_{\epsilon}(X(t_j)- X(t_{j-1}))\,dt_1\cdots\,dt_n \,d\mu(x)\] 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 self-intersection local time of \(X\) with respect to the measure \(\mu\) is defined as \[\gamma_{n}(\mu)=\lim_{\epsilon\to 0}\,\sum_{k=0}^{n-1}(-1)^{k} {n-1 \choose k}(u^1_{\epsilon}(0))^{k} \alpha_{n-k,\epsilon}(\mu,\lambda)\] where \(u^1_{\epsilon}(x)\overset{\text{def}}{=} \int f_{\epsilon}(x-y)u^1(y)\,dy\), 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 \(\{\gamma_{n}(\mu_x),\,x\in R^m\}\).


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 self-intersection local times
  • Continuity
  • Stable mixtures
  • Examples
  • A large deviation result
  • Appendix A. Necessary conditions
  • Appendix B. The case \(n=3\)
  • Bibliography
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