We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator $M(t)$ can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection $\Pi$ on a subspace of sufficiently regular functions. Then the eigenfunctions of $M(t)$ with small eigenvalues have only a very small component in the image of $\Pi$."
We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced by Hairer and Mattingly in Ann. of Math. (2) 164 (2006).
One of the main novel technical tools is an almost sure bound from below on the size of "Wiener polynomials," where the coefficients are possibly non-adapted stochastic processes satisfying a Lipschitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris' lemma, which is unavailable in the present context.
We conclude by showing that the two-dimensional stochastic Navier-Stokes equations and a large class of reaction-diffusion equations fit the framework of our theory.
"A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs." Electron. J. Probab. 16 658 - 738, 2011. https://doi.org/10.1214/EJP.v16-875