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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
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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 \(\{\gamma_{n}(\mu_x),\,x\in R^m\}\).

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 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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