Math @ Duke

Publications [#243865] of Jonathan C. Mattingly
search arxiv.org.Papers Published
 Hairer, M; Mattingly, JC, Ergodicity of the 2D NavierStokes equations with degenerate stochastic forcing,
Annals of Mathematics, vol. 164 no. 3
(2006),
pp. 9931032, Annals of Mathematics, Princeton U, ISSN 0003486X [math.PR/0406087], [doi]
(last updated on 2021/05/14)
Abstract: The stochastic 2D NavierStokes 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 ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in L02(double struck T sighn2). Unlike previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property, introduced in this work, and an approximate integration by parts formula. The first, when combined with a weak type of irreducibility, is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds under a HÃ¶rmandertype condition. This requires some interesting nonadapted stochastic analysis.


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