SUBLAPLACIANS WITH DRIFT
25
PROOF.
By proposition 1.6.2, there is 6 0 and c$ 0 such that
ps(x,y)dy 5, 0 s r2
JBC
for all r 1 and x G G.
Let us fix e 0. Then, by proposition 1.6.3, there is c c$ such that
pri(x,y)dy sS
JB,
lBcr{x)c
for all r 1 and x G G.
We have
e5 I pri(x,y)dy
JBcr(x)c
=Px[z(r2) G Bcr(x)c]
EP* [P (r2  rfcr, *(7fcr), £
c r
(*) c ) i{r?crr»}]
£
p
 [P
(r2
 r2*cr, z(r2*cr), £
c r
( ^
c r
) ) ) i{Tfcr r2}]
= B P  [P (r2  r
c r
, e, Bcr(e)) i
{ r

c r r a }
]
5P*[r2*crr2]
and hence
P * K
r
r2]
£
which proves the lemma.
PROOF OF LEMMA 3.1.1. Let t0 G (0, \a~2r2) and let us denote by Ato the
section
Ato =An{t0} x Sr(e).
Let c 1 + a and let r^r be as in the lemma 3.2.2.
Let also u 0 satisfying u(s,y) 1, for (s,y) G A and (J^ + L) u = 0 in
(0,a2r2) x B2cr(e). Then
w(t, a:) £
p
* [u (t0, z(t  t
0
));
rcxr
tt0]
( 3
^
3 )
£ p * [iAt0 (*((*  *o)); TZ, t 1
0
]
= / ptt0(x,y)dy  Px[r*r t  t0]
for all (t,z) G (a~
2
r
2
,a
2
r
2
) x J5ar(e).
Now, by lemma 3.2.1, there is 5 0 and £o G (0,1) such that
/ ptt0(x,y)dy25
for all (t,x) G (a 2r2,a2r2) x Bar(e), if Ato satisfies
(3.2.4) \Ato\ £0\Br(e)\.
If we assume that \A\ £(0, r2) x i?r(e), with £ G [£o, 1) near enough to 1,
then A will always have a section At0 with to G (0, ^a
_ 2
r
2
) and satisfying (3.2.4).