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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)

- Martin Hairer, Leonid Koralov, Zsolt Pajor-Gyulai
- 2014

We consider a two-parameter averaging-homogenization type elliptic problem together with the stochastic representation of the solution. A limit theorem is derived for the corresponding diffusion process and a precise description of the two-parameter limit behavior for the solution of the PDE is obtained.

This article is devoted to the numerical study of various finite difference approximations to the stochastic Burgers equation. Of particular interest in the one-dimensional case is the situation where the driving noise is white both in space and in time. We demonstrate that in this case, different finite difference schemes converge to different limiting… (More)

- Martin Hairer
- 2005

We consider a weak form of controllability for system that have a conserved quantity and satisfy a condition of Hörmander type. It is shown that such systems are approximately controllable under a weak growth condition for the conserved quantity. The proof of the result combines analytic tools with probabilistic arguments. A counterexample is given that… (More)

- Martin Hairer
- 2003

We study the ergodic properties of finite-dimensional systems of SDEs driven by non-degenerate additive fractional Brownian motion with arbitrary Hurst parameter H ∈ (0, 1). A general framework is constructed to make precise the notions of " invariant measure " and " stationary state " for such a system. We then prove under rather weak dissipativity… (More)

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