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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

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.

• 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