1. Preliminaries.
The letter c with subscripts will denote positive finite constants whose values
are unimportant. We begin numbering the constants anew in each proposition,
theorem, lemma, and corollary. We use "Bes(3)" as an abbreviation for "Bessel
process of index 3" and "BM x Bes(3)" for the planar process (Xt, Yt) where Xt is
a Brownian motion and Yt is a Bes(3) independent of Xt. We will use the symbol
to denote the distribution of a process starting from x when it is clear from
the context which process we are talking about. When this notation might be
confusing, we will write
and Fy to denote the distributions of
a Brownian motion (one- or two-dimensional), a Bes(3), a BM x Bes(3), and the
process V defined in (2.3) below, respectively. Similar subscript conventions will
apply to expectations.
For Borel sets A we define
rA =
(A) = inf{* : Zt £ A}, TA = T(A) = inf{t: Zt A},
where Zt is a process such as planar Brownian motion or a BM x Bes(3). We will
use J5(x,r) to denote the open ball of radius r about x. We use dA to denote the
boundary of a Borel set A.
Throughout this paper we will make frequent use of /i-path transforms.
Recall that if
is a strong Markov process killed on exiting a domain D
with transition densities p(t,x,y) and h is positive and
in D, then
h(y)p(tJx^y)/h(x) are densities of a strong Markov process (P^,Xt) in D. For de-
tails and results concerning /i-path transforms, see Doob (1984) or Bass (1995).
The result we will use most frequently is that
E%Y = W[h(XT)Y\/h{x) (1.1)
if T is a stopping time and Y is .^-measurable and either bounded or nonnegative.
It is well known that if one takes one dimensional Brownian motion killed
on hitting (—oo,0] and /i-path transforms it by the harmonic function h(x) = x,
one gets a Bes(3).
We will use several times a bound on the probability that 2-dimensional
Brownian motion exits a rectangle through the vertical sides. If Zt = (Xt,Yt) is
2-dimensional Brownian motion and DR = {(x,y) :0 y 1, |x| # } , then there
exist constants c\ and C2 such that
P ^ ) ( | X
T ( D R )
| =R)
Cle~c R.
A proof of this may be found in Bass (1995), p. 180.
The estimate of (1.2) also holds when Zt is a BM x Bes(3). To see this, we
use the fact that Yt is equal in law to the modulus of a 3-dimensional Brownian
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