  In this paper, we derive global sharp heat kernel estimates for symmetric
alpha-stable processes (or equivalently, for the fractional Laplacian with zero
exterior condition) in two classes of unbounded C^{1,1} open sets in R^d:
half-space-like open sets and exterior open sets. These open sets can be
disconnected. We focus in particular on explicit estimates for p_D(t,x,y) for
all t>0 and x, y\in D. Our approach is based on the idea that for x and y in
$D$ far from the boundary and t sufficiently large, we can compare p_D(t,x,y)
to the heat kernel in a well understood open set: either a half-space or R^d;
while for the general case we can reduce them to the above case by pushing $x$
and $y$ inside away from the boundary. As a consequence, sharp Green functions
estimates are obtained for the Dirichlet fractional Laplacian in these two
types of open sets. Global sharp heat kernel estimates and Green function
estimates are also obtained for censored stable processes (or equivalently, for
regional fractional Laplacian) in exterior open sets.
