Attractivity, invariance and ergodicity for SDEs on Riemannian manifolds
  We give a sufficient condition on nonlinearities of an SDE on a compact connected Riemannian manifold $M$ which implies that laws of all solutions converge weakly to the normalized Riemannian volume measure on $M$. This result is further applied to characterize invariant and ergodic measures for various SDEs on manifolds. 
Lubomir Banas, Zdzislaw Brzezniak, Martin Ondrejat, Andreas Prohl
math.PR math.NA