English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We present a theory of hypoellipticity and unique ergodicity for semilinearparabolic stochastic PDEs with "polynomial" nonlinearities and additive noise,considered as abstract evolution equations in some Hilbert space. It is shownthat if Hormander's bracket condition holds at every point of this Hilbertspace, then a lower bound on the Malliavin covariance operator $M_t$ can beobtained. Informally, this bound can be read as "Fix any finite-dimensionalprojection $\Pi$ on a subspace of sufficiently regular functions. Then theeigenfunctions of $M_t$ with small eigenvalues have only a very small componentin the image of $\Pi$".  We also show how to use a priori bounds on the solutions to the equation toobtain good control on the dependency of the bounds on the Malliavin matrix onthe initial condition. These bounds are sufficient in many cases to obtain theasymptotic strong Feller property introduced in [HairerMattingly06].  One of the main novel technical tools is an almost sure bound from below onthe size of "Wiener polynomials", where the coefficients are possiblynon-adapted stochastic processes satisfying a Lipschitz condition. Byexploiting the polynomial structure of the equations, this result can be usedto replace Norris' lemma, which is unavailable in the present context.  We conclude by showing that the two-dimensional stochastic Navier-Stokesequations and a large class of reaction-diffusion equations fit the frameworkof our theory.

A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs