By Fengyu WANG, Beijing Normal University

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.