In this paper, we prove the existence of a unique strong solution to a
stochastic tamed 3D Navier-Stokes equation in the whole space as well as in the
periodic boundary case. Then, we also study the Feller property of solutions,
and prove the existence of invariant measures for the corresponding Feller
semigroup in the case of periodic conditions. Moreover, in the case of periodic
boundary and degenerated additive noise, using the notion of asymptotic strong
Feller property proposed by Hairer and Mattingly \cite{Ha-Ma}, we prove the
uniqueness of invariant measures for the corresponding transition semigroup.Comment: 38Pages, Correct some error
