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
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