  It has been suggested in 1999 that a certain volume growth condition for
geodesically complete Riemannian manifolds might imply that the manifold is
stochastically complete. This is motivated by a large class of examples and by
a known analogous criterion for recurrence of Brownian motion. We show that the
suggested implication is not true in general. We also give counter-examples to
a converse implication.
