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.