Almost-sure exponential mixing of passive scalars by the stochastic Navier-Stokes equations

We deduce almost-sure exponentially fast mixing of passive scalars advected by solutions of the stochastically-forced 2D Navier-Stokes equations and 3D hyper-viscous Navier-Stokes equations in $\mathbb T^d$ subjected to non-denegenerate $H^\sigma$-regular noise for any $\sigma$ sufficiently large. That is, for all $s > 0$ there is a deterministic exponential decay rate such that all mean-zero $H^s$ passive scalars decay in $H^{-s}$ at this same rate with probability one. This is equivalent to what is known as \emph{quenched correlation decay} for the Lagrangian flow in the dynamical systems literature. This is a follow-up to our previous work, which establishes a positive Lyapunov exponent for the Lagrangian flow-- in general, almost-sure exponential mixing is much stronger than this. Our methods also apply to velocity fields evolving according to finite-dimensional fluid models, for example Galerkin truncations of Navier-Stokes or the Stokes equations with very degenerate forcing. For all $0 \leq k < \infty $ we exhibit many examples of $C^k_t C^\infty_x$ random velocity fields that are almost-sure exponentially fast mixers.