Brownian motion and harnack inequality for ... - Math.caltech.edu

234 M. AIZENMAN AND B. SIMON
which tends to zero as f&O, by Lemma 4.4. On the other hand,
Since If(z)l S (const) g(z) in the
where f is the integral kernel of (- +A + I)-'.
region )zI 5 i (a restriction needed if Y = 2), this term goes to zero as a&O. These
two estimates imply (4.4).
For Y = I , let V E K,,
Moreover,
which vanishes as tL0.
To prove the converse, let P,T(x - y) be the integral kernel of exp{ - sH,} and
suppose that Y 2 3. We claim that for Ix -yI 5 tl12 we have
Postponing the proof of (4.5), we note that it implies that, for a sufficiently small,
thus (4.4) implies V E K,,.
Since P,,(x -y) = (2ns)-"/2exp( -(x -yI2/2s), we see that
if Ix - yJ 5 f'12. This proves the required result (4.5).