Browsing by Author "Hairer, Martin"

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Open Access
A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs
(Electronic Journal of Probability, 2011-05-09) Hairer, Martin; Mattingly, Jonathan C
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.
Open Access
Ergodic properties of highly degenerate 2D stochastic Navier-Stokes equations
(Comptes Rendus Mathématique. Académie des Sciences. Paris, 2004) Hairer, Martin; Mattingly, Jonathan C
This Note presents the results from "Ergodicity of the degenerate stochastic 2D Navier-Stokes equation"; by M. Hairer and J.C. Mattingly. We study the Navier-Stokes equation on the two-dimensional torus when forced by a finite dimensional Gaussian white noise and give conditions under which the system is ergodic. In particular, our results hold for specific choices of four-dimensional Gaussian white noise. © 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Open Access
Spectral gaps in wasserstein distances and the 2d stochastic navier-stokes equations
(Annals of Probability, 2008-11-01) Hairer, Martin; Mattingly, Jonathan C
We develop a general method to prove the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł p-type norm, but involves the derivative of the observable as well and hence can be seen as a type of 1-Wasserstein distance. This turns out to be a suitable approach for infinite-dimensional spaces where the usual Harris or Doeblin conditions, which are geared toward total variation convergence, often fail to hold. In the first part of this paper, we consider semigroups that have uniform behavior which one can view as the analog of Doeblin's condition. We then proceed to study situations where the behavior is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the two-dimensional stochastic Navier-Stokes equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show that the stochastic Navier-Stokes equations' invariant measures depend continuously on the viscosity and the structure of the forcing. © Institute of Mathematical Statistics, 2008.
Open Access
Yet another look at Harris’ ergodic theorem for Markov chains
(2011) Hairer, Martin; Mattingly, Jonathan C