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The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is er-godic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in… (More)

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… (More)

- M. Hairer
- 2013

- M. Hairer, A. M. Stuart, J. Voss
- 2005

Data assimilation is formulated in a Bayesian context. This leads to a sampling problem in the space of continuous time paths. By writing down a density in path space, and conditioning on observations, it is possible to define a range of Markov Chain Monte Carlo (MCMC) methods which sample from the desired distribution in path space, and thereby solve the… (More)

There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris' theorem. The first uses the… (More)

- J.-P. Eckmann, M. Hairer
- 2001

We consider the stochastic Ginzburg-Landau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The low-lying frequencies are then only connected to this forcing through the non-linear (cubic) term of the Ginzburg-Landau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant… (More)

- J.-P. Eckmann, M. Hairer
- 1999

We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, Rey-Bellet [EPR99a,EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is… (More)

- M. Hairer, A. M. Stuart, J. Voss
- 2006

In many applications it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite dimensional analogue of the Lan-gevin SDE used in finite dimensional sampling. In this paper nonlinear SDEs, leading to nonlinear SPDEs for… (More)

- M. Hairer
- 2001

We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions amount essentially to the fact that the equation transmits the noise to all its determining modes. Several examples are… (More)

- J.-P. Eckmann, M. Hairer
- 2002

We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = m i=1 X T i X i + X0 + f , where the Xj denote first order differential operators, f is a function with at most polynomial growth, and X T i denotes the formal adjoint of Xi in L 2. For any ε > 0 we show that an inequality of the form u δ,δ ≤ C(u0,ε + (K +… (More)