A Dynamics-Informed Gaussian Process Framework for 2D Stochastic Navier-Stokes via Quasi-Gaussianity

Yet a fundamental gap remains: while these methods depend critically on the choice of prior covariance kernel, most kernels are selected for computational convenience (e.g., Gaussian/RBF kernels) or generic smoothness assumptions (e.g., Mat ern) rather than being rigorously grounded in the system's long-time statistical structure. Recent breakthroughs in stochastic PDE theory now make it possible to close this gap, constructing priors directly from the invariant-measure geometry of the underlying dynamics. Recent work of Coe, Hairer, and Tolomeo [7] establishes a remarkable geometric property of the two-dimensional stochastic Navier-Stokes (2D SNS) equations: although the dynamics are highly nonlinear, their unique invariant measure is equivalent-in the sense of mutual absolute continuity-to the Gaussian invariant measure of the linearized Ornstein-Uhlenbeck (OU) process. Equivalence means the two measures share the same support, null sets, and typical events, differing only by a positive Radon-Nikodym derivative. This reveals that the equilibrium statistical geometry is Gaussian, even when individual realizations are not.