  A unified treatment is given of some results of H. Donnelly-P. Li and L.
Schwartz concerning the behaviour of heat semigroups on open manifolds with
given compactifications, on one hand, and the relationship with the behaviour
at infinity of solutions of related stochastic differential equations on the
other. A principal tool is the use of certain covers of the manifold: which
also gives a non-explosion test. As a corollary we obtain known results about
the behaviour of Brownian motions on a complete Riemannian manifold with Ricci
curvature decaying at most quadratically in the distance function.
