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A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs

dc.contributor.author Hairer, M
dc.contributor.author Mattingly, Jonathan Christopher
dc.date.accessioned 2015-03-20T17:55:04Z
dc.date.issued 2011-05-09
dc.identifier.uri http://hdl.handle.net/10161/9521
dc.description.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 operatorμt can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection on a subspace of sufficiently regular functions. Then the eigenfunctions of μt with small eigenvalues have only a very small component in the image of Π." 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 in [HM06]. 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 Lips chitz 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.
dc.relation.ispartof Electronic Journal of Probability
dc.title A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs
dc.type Journal article
pubs.begin-page 658
pubs.end-page 738
pubs.organisational-group Duke
pubs.organisational-group Mathematics
pubs.organisational-group Statistical Science
pubs.organisational-group Trinity College of Arts & Sciences
pubs.publication-status Published
pubs.volume 16
dc.identifier.eissn 1083-6489


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