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: