CHAPTER 1
Introduction
We study the continuity of renormahzed self-intersection local times for the
multiple intersections of a large class of strongly symmetric Levy processes in i?
m
,
m 1, 2, including symmetric stable processes and many processes in their domains
of attraction. We do this by comparing these processes to Wick power Gaussian
chaos processes using an isomorphism theorem which generalizes an isomorphism
theorem of Dynkin.
Intersection local times "measure" the amount of self-intersections of a stochas-
tic process, say, X(t) G
Rm.
To define the n—fold self-intersection local time, the
natural approach is to set
c w ( M )
d=
(Li)
I t fe{X(h) -X)f[ fe(X(tj) ~ X(^-l) ) dh dtn dfJL(x)
J J{0t1---tnt}
J = 2
where fe is an approximate 6—function at zero, and take the limit as e —• 0.
Intuitively, this gives a measure of the set of times (ti,... , tn) such that
X{t1) = -.. = X{tn) = x,
where the "n-multiple points" x G Rm are weighted by the measure \i. However,
in general, this limit does not exist because of the effect of the integral in the
neighborhood of the diagonal. The method used to compensate for this is called
renormalization. One subtracts from an,e(/i,£) terms involving lower order inter-
sections ak,e(fJ,t) for k n, in such a way that a finite limit results. This was
originally done by Varadhan [27] for double intersections of Brownian motion in
the plane with ji taken to be Lebesgue measure. Varadhan's work stimulated a large
body of research which is summarized by Dynkin in [6]. Renormahzed intersection
local times have turned out to be the right tool for the solution of certain "classical"
problems such as the asymptotic expansion of the area of the Wiener and stable
sausage in the plane and fluctuations of the range of stable random walks. (See Le
Gall [10, 9], Le Gall-Rosen [12] and Rosen [25]). For a clear account of research
on Brownian intersection local times up to 1990 see Le Gall's lecture notes [11].
For more recent results see Bass and Khoshnevisan [2] and Rosen [24].
This research of both authors was supported, in part, by grants from the National Science Foundation and PSC-
CUNY. The research of Jay Rosen was also supported, in part, by the U.S.-Israel Binational Science Foundation.
Received by the editor January 13, 1997; and in revised form May 5, 1998.
Previous Page Next Page