2. HISTORY 7

estimates of the type

P(Rn ≥ ϕ(n)) or P(Rn ≤ ϕ(n))

for functions ϕ(n) that grow relative quickly. By contrast, in our paper we look at

P(Rn − ERn ≥ ϕ(n))

for functions ϕ(n) that grow not quite so quickly. The paper [12] determines

lim

n→∞

nd/(d+α)

log

Ee−λRn

,

when X1 is in the domain of attraction of a symmetric stable process of index α.

This can be used to obtain information on P(Rn ≤ ϕ(n)). It is shown in [17] that

lim

n→∞

−

1

n

log P(Rn ≥ nx)

exists, while [16] examines

lim

n→∞

1

n

log

EeλRn

.

Related to our paper is one by Chen [8], which looks at moderate deviations

of the joint range Jn, the cardinality of the set of points that are each visited by

each of p independent random walks.

We will be much briefer concerning the literature of intersection local times.

There are a large number of papers, but we only mention those relevant to this pa-

per. Given two independent Brownian motions Vt, Wt in two dimensions, formally

the intersection local time is the quantity

t

0

t

0

δ0(Vs − Wu) du ds,

which is a measure of how often the two Brownian motions intersect each other;

here δ0 is the delta function. To give a meaning to this, we define the intersection

local time by

(2.8) lim

ε→0

t

0

t

0

pε(Vs − Wu) ds du,

where pε is a suitable approximation to the identity, for example, the density of

a two-dimensional Brownian motion at time ε. If one now wants to define self-

intersection local time of one planar Brownian motion, one cannot simply replace

Vs by Ws. If one does, one gets an identically infinite random process. Varadhan

[30] proved that provided one renormalizes properly, one can get a finite limit.

There are a number of renormalizations possible. We use the following: let

(2.9) γε(t) =

t

0

u

0

pε(Ws − Wu) ds du,

and let

(2.10) γt = lim[γε(t)

ε→0

− Eγε(t)].

The limit exists almost surely, and is called renormalized self-intersection local time

for planar Brownian motion.

LeGall [26] proved that

(2.11)

Eeλγ1