2

1. INTRODUCTION

For Brownian motion in the plane and with /i taken to be Lebesgue measure,

Dynkin [5] introduced the idea of studying an,e(/i, A), where A is an exponential

random variable with mean one, independent of X(t), and showed how this random-

ization of time leads to technical simplifications. Also, he introduced the following

renormalization formula

7„(M) =' lim E(- 1 )' C ( n I ^ ( O ^ - U / M ) (1.2)

where u\(x) = J fe(x — y)u1(y)dy and u1(a;) = j

0

e~tpt(x) dt is the 1-potential

density of X(t).

Let

7»,e(M) = £ ( - ! ) * ( " . j K H O ^ n - ^ ^ A ) . (1.3)

k=0 ^ '

Heuristically, one may think of 7n,e(/0

a s

being equal to

/I

/

e

(X(ti)-a; ) (1.4)

{0ti---t

n

A}

n

J ] {/c(Xfo) - Xfo-i)) - (5(t, - tj-JuKO)} dh • • • *

n

d/x(x).

In this formulation the ^-functions compensate for the singularities that occur when

various of the U are close to each other.

1. Statement of results

In this paper we consider renormalized self-intersection local times 7n(/-0 f°r &

large class of radially symmetric Levy processes in

Rm,

m — 1, 2 and positive finite

measures n on

R171.

We define /ix(*) = M

x

+ ') to be the measure \x translated by

x G

R171

and study the continuity of the stochastic process

{7nW,xer}. (1.5)

Let (ft, .F(£), X(£), P x ) be a radially symmetric Levy processes in P m , m — 1, 2

with 1-potential density

it1(x,y).

When

tt1(x,x)

oo, X(t) has a local time. In

this case the intersection local time of X(t) can be expressed as a simple function

of the local time. We do not deal with this case in this memoir, but only consider

Levy processes for which ul{x,x) — oo.

Clearly v}{x^y) =

ul(x

— y,0) and since X(t) is radially symmetric we some-

times write these terms as ul(x — y) or ul(\x — y\). The results obtained in this

paper are valid for a large class of radially symmetric Levy processes which we

say are in Class A. This class contains the symmetric stable processes and many