Stochastic Tamed Navier–Stokes Equations on R³: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

Röckner and Zhang (Probab Theory Relat Fields 145, 211–267, 2009) proved the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space and for the periodic boundary case using a result from Stroock and Varadhan (Multidimensional diffusion processes, Springer, Berlin, 1979). In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the L⁴ L 4 -norm of the solution from the torus to R³ R 3 , see Lemma 5.1 and thus establish the existence of an invariant measure on R³ R 3 for a time-homogeneous damped tamed 3D Navier–Stokes equation, given by (6.1).