Michael B Marcus 
Renormalized Self-Intersection Local Times and Wick Power Chaos Processes [PDF ebook] 

Sufficient conditions are obtained for the continuity of renormalized self-intersection local times for the multiple intersections of a large class of strongly symmetric Levy 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 Levy 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 $/{/gamma_{n}(/mu_x), /, x/in R^m/}$.