1. STATEMENT OF RESULTS
5
(See Remark 3.1 at the end of Chapter 3 for some comments on the Fourier trans-
form of (ux( ))2n).
Here is a concrete application of Corollary 1.1.
COROLLARY
1.2. Let X = {X(t),t G R+} be a Levy process in Rm, m = 1,2,
in Class A and {^n{Hx), x G
R171}
be the n-fold renormalized self-intersection local
time process of X, with respect to a finite positive measure \i on R™. If either
/ JM^ (logx)71'1 dx oo (1.13
h
x
or
(u\x -
y))2n{\og
\/\x -
y\Yn+V
dfi(x) d/xfe) oo (1.14)
\x-y\e
for any e,6 0, then \i G Q2n and {^n{l^x), % £ R171} is continuous P% almost
surely for all y G
R771.
SL
\m\=0[n_
e
) as 1^1-00. (1.15)
In particular, for Brownian motion in
R2,
(1.13) holds when
1
,(log|^)2
Furthermore for a Levy process X in R2 in Class A with Levy exponent asymp-
totic to
A2/(log|A|)a,
a 0, as A - oo, (see (1.19) and (1.21) below), the n-fold
renormalized self-intersection local time process of X, with respect to a positive
measure \i on R2, is continuous almost surely if (1.15) holds with 2n replaced by
2n(l + a/2).
We do not know whether (1.11) is a necessary condition for continuity of the
2n-th Wick power chaos associated with
u1
for any measure /i. Based on Theorem
1.5, [16] and the results in [21] we suspect that at least for a class of smooth
measures /i, a necessary and sufficient condition for continuity of the 2n-th Wick
power chaos associated with
w1(-)
is the one in (1.11) but with n replaced by 1/2.
We do not know how to prove this. The methods of [21], which prove a result of
this nature for second order Wick power chaos processes, do not extend to higher
order Wick power chaos processes.
The isomorphism theorems we develop can be used to obtain other path prop-
erties of renormalized self-intersection local times besides continuity. In [17] and
[16] we used Dynkin's isomorphism theorem to obtain moduli of continuity re-
sults for continuous additive functionals of Levy processes. We can do the same
here for {^n(iix),x G
R171}.
However, since there is little new involved we will
leave this to the interested reader. Instead, we demonstrate the power of the iso-
morphism theorem approach by obtaining a bound on the exponential moment of
suPxG[-i,i]m l7n(Mx)|1/^n- The following theorem is proved in Chapter 9:
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