Abstract
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.
Highlights
The overarching goal of this article is to prove the unique ergodicity of a class of nonlinear stochastic partial differential equations (SPDEs) driven by a finite number of Wiener processes
The remainder of this section is devoted to a short discussion of the main techniques used in the proof of such a result and in particular on how to obtain a bound of the type 1.2 for a parabolic stochastic PDE
It follows from the theory of Malliavin calculus, see for example [Mal[97], Nua95] that, for any Hilbert space H, there exists a closed unbounded linear operator D : L2(Ω, R) ⊗ H → L2ad(Ω, Ft, CM ) ⊗ H such that DΦt coincides with the object described above whenever Φt is the solution map to (3.1)
Summary
The overarching goal of this article is to prove the unique ergodicity of a class of nonlinear stochastic partial differential equations (SPDEs) driven by a finite number of Wiener processes. We prove the nondegeneracy of the Malliavin covariance matrix under an assumption that the linear span of the successive Lie brackets[1] of vector fields associated to N and the gk is dense in the ambient (Hilbert) space at each point This is very reminiscent of the condition in the “weak” version of Hormander’s “sum of squares” theorem. Bounds on the norm of the inverse of the Malliavin matrix are the critical ingredient in proving ergodic theorems for diffusions which are only hypoelliptic rather than uniformly elliptic. This shows that the system has a smooth density with respect to Lebesgue measure. There are many natural infinite dimensional Markov processes whose transition probabilities do not converge in total variation to the system’s unique invariant measure. (See examples 3.14 and 3.15 from [HM06] for more discussion of this point.) In these settings, this fact precludes the use of “minorization” conditions such as infx∈C Pt(x, · ) ≥ cν( · ) for some fixed probability measure ν and “small set” C. (see [MT93, GM06] for more and examples were this can be used.)
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