# Renormalized Self-Intersection Local Times and Wick Power Chaos Processes

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*Michael B. Marcus; Jay Rosen*

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

#### Table of Contents

# Table of Contents

## Renormalized Self-Intersection Local Times and Wick Power Chaos Processes

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. Wick products 1017
- Chapter 3. Wick power chaos processes 1421
- Chapter 4. Isomorphism theorems 2936
- Chapter 5. Equivalence of two versions of renormalized self-intersection local times 5360
- Chapter 6. Continuity 7481
- Chapter 7. Stable mixtures 7784
- Chapter 8. Examples 8491
- Chapter 9. A large deviation result 8693
- Appendix A. Necessary conditions 8996
- Appendix B. The case n = 3 103110
- Bibliography 124131