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.
This work was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. T.K. acknowledges the support of the National Science Centre: NCN grant 2016/23/B/ST1/00492. S.P. acknowledges the support of the National Science Center grant 2017/25/B/ST1/02584.
The authors would like to thank an anonymous referee and the Associate Editor for careful reading of the paper and for providing many useful remarks and comments which have lead to a substantially improved and more complete presentation. The authors would like to thank the Banach centre for its hospitality.
"Ergodicity for stochastic equations of Navier–Stokes type." Electron. Commun. Probab. 27 1 - 10, 2022. https://doi.org/10.1214/21-ECP443