6 RICHARD F. BASS AND KRZYSZTOF BURDZY
motion, and the probability that 4-dimensional Brownian motion exits the cylinder
{(xi,X2,X3,X4) : |#i| R,x\ + x\ + x\ 1} through the ends is bounded by the
right hand side of (1.2) by Bass (1995), p. 180.
If Zt is a 2-dimensional Brownian motion and H is the upper half plane,
then there exists c\ such that
sup
P(x'y)(ZT(B(0,2)nH)
i OH) Cly; (1.3)
(x,y)eB(0,l)r\H
see Bass (1995), p. 313, for a proof.
We will use several times the boundary Harnack principle; see Bass(1995),
p. 178.
Theorem 1.1. Let D be a connected Lipschitz domain, XQ D, V open, and K
compact so that K C V. There exists c\ depending on K,V, and D such that if u
and v are two positive harmonic functions on D vanishing continuously on V n dD,
then
U(x) U{X0)
—7-TCI— r, XeKClD.
v(x) v{xo)
Note that if rA = {rx : x G A} for any set A, then by scaling the constant
c\ for the triple (rK,rV,rD) can be taken to be independent of r.
Previous Page Next Page