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

260 M. AIZENMAN AND B. SIMON
Thus, (A.2.10) follows from the one-dimensional case (Lemma A.2.7) and the
bound
which follows from the monotonicity of S'/~P,'')(X, y) in s.
THEOREM A.2.8.
(A.2.11) and let
Let 52 be a rounded convex set in R'. Let P, be given by
Qs(x9 u) = +(x)-1WV'(x9 Y)
PL2 being the integral kernel of the semigroup generated by the Dirichlet Laplacian
in 52. Then (A.2.10) holds for any c > 0. Moreover, C, depends onh on Y and on the
ratio
sup[ dist(x, aQ)+(x)]/inf[ dist(x, aQ)+(x)].
In particular, the Same C, will work for all balls.
Proof: We bound instead
(A.2.12) dist(x,aS2)-1P:(x, y)dist(y, an)
by C, P, I +,)s(x, y) independently of 52 and then use Lemma A.2.5. Given x, pick
z E a52 so that dist(z,x) = dist(x, 352) and let T be the half-space with z E aT, and
Cl C T. By translation and rotation, we can suppose without loss of generality
that T = {zlzl > 0). Then dist(x, aS2) = xI and dist(y, a52) Sy,. Since PR is
monotone in 52, we see that (A.2.12) is bounded by
so Lemma A.2.7 yields the required bound.
Appendix 3. Some Remarks on First Exit Times
Let Cl be an open subset of R' and let T be the first hitting time for R'\52. The
first fact that we want to note helps explain why Chung-Rau [7] obtain results
which depend only on )QI.
THEOREM A.3. I .
Let f be a monotone non-decreasing function on [O,m). Then