Abstract
Invariant Measures for Three-Dimensional Stochastic Primitive Equations with Horizontally Dependent Transport Noise
We study invariant measures for three-dimensional stochastic primitive-equation models driven by projected horizontal Stratonovich transport noise. We assume that the transport field is horizontal, independent of the vertical variable, and horizontally divergence-free. Under this structure the projected noise is skew-adjoint in the hydrostatic $L^2$ space; after taking the baroclinic projection or one vertical derivative, the stochastic hydrostatic pressure disappears and the transport becomes an exact divergence-free derivation. Consequently the high-power baroclinic and vertical-derivative balances have exact Stratonovich cancellations. Adapting the logarithmic strong-moment mechanism of Glatt-Holtz--Kukavica--Vicol--Ziane, we obtain a time-averaged logarithmic $H^2$ bound. A localized continuous-dependence estimate yields the Feller property on the mean-zero $H^1$ phase space, and the Krylov--Bogoliubov procedure then gives an invariant probability measure.