A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs

By Martin Hairer, Jonathan C. Mattingly. 2008

167 citations recorded by the source corpus.

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 Hormander'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.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/515f09c811abd4df43029d92d567601368510df7

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Citations in the corpus

Convex integration and phenomenologies in turbulence (2019, background): “[94, 95, 122] for two dimensional flows and [2, 80] for dyadic shell models (see also [53, 90]).”
Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths (2011, result): “ of this section is to argue that while the probabilistic intuition described above is certainly correct, one can have a much more pathwise perspective on Norris’s lemma. This was already apparent in [21], REGULARITY OF LAWS 9 where the authors obtain a result that is similar in ﬂavor to Norris’s le”
Smoothness of the density for solutions to Gaussian rough differential equations (2012, unknown): “s less than 1, we have Z· 0 bs ds ∞ ≤Mkzk(1−θ/(2γ))N−1(1−β/γ) ∞ R κ. 26 CASS, HAIRER, LITTERER AND TINDEL Once again, we use an interpolation inequality to strengthen this bound. Indeed, we have (see [21], Lemma 6.14, for further details) k∂tfk ∞≤Mkfk∞max 1 T ,kfk−1/(γ+1)k∂ tfk 1/(γ+1) γ , and app”
On Malliavinʼs proof of Hörmanderʼs theorem (2011, methodology): “ll always consider functions on the time interval [0,1]. Before we turn to the proof of this result, we introduce a very useful notation which, to the best of the author’s knowledge, was ﬁrst used in [HM11]. Given a family A = {Aε}ε∈(0,1] of events depending on some parameter ε > 0, we say APPLIC”
Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction (2013, background): “Their approach have been generalized to more general equations in [40].”