  By using the local dimension-free Harnack inequality established on
incomplete Riemannian manifolds, integrability conditions on the coefficients
are presented for SDEs to imply the non-explosion of solutions as well as the
existence, uniqueness and regularity estimates of invariant probability
measures. These conditions include a class of drifts unbounded on compact
domains such that the usual Lyapunov conditions can not be verified. The main
results are extended to second order differential operators on Hilbert spaces
and semi-linear SPDEs.
