Ergodicity for stochastic equation of Navier-Stokes type

In this note we consider a simple example of a finite dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic Navier-Stokes Equations (NSEs), and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant probability measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant probability measure, there exist another one. In particular, the generator of the system is not hypoelliptic