Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing

By Martin Hairer, Jonathan C. Mattingly. 2004

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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 ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in L 2 0 (T 2 ). 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 un

Subfields: Applied Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/c0a4750f227881d034ed23221822d451fefa5f46

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Citations in the corpus

Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations (2009, background): “While even simple, formally elliptic, linear SPDEs can have transition probabilities which are mutually singular, the bulk of recent work has been motivated by equations driven by noise which is “degenerate” to varying degrees [EH01, BKL01, EMS01, KS02, Mat02b, HM06 , HM08a].”
Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator (2012, methodology): “For that purpose, we extend the delicate argument introduced by Henderson et al. (2011) and based on the theory of asymptotically strong Feller Markov chains (see Hairer and Mattingly (2006)).”
A moderate deviation principle for 2-D stochastic Navier-Stokes equations (2015, methodology): “The ergodic properties and invariant measures of the 2-D stochastic Navier–Stokes equations were studied in [10] and [14].”
Densities for rough differential equations under Hormander's condition (2007, background): “Examples of driving signals include fractional Brownian motion with Hurst parameter H > 1/4, the Brownian Bridge returning to zero after time T and the Ornstein-Uhlenbeck process.”
Convex integration and phenomenologies in turbulence (2019, background): “[94, 95, 122] for two dimensional flows and [2, 80] for dyadic shell models (see also [53, 90]).”
Local Lipschitz Continuity in the Initial Value and Strong Completeness for Nonlinear Stochastic Differential Equations (2013, background): “Lemma 4.10 in Hairer & Mattingly [15] also includes an inequality similar to (14) in the case where U1 is the squared Hilbert space norm for the vorticity formulation of the two-dimensional stochastic Navier-Stokes equations.”
A Gentle Stochastic Thermostat for Molecular Dynamics (2009, methodology): “We state a basic result which relates ergodicity to regularity of solutions of the Fokker-Planck equation; for similar results see [8, 18, 19], as well as earlier works on ergodicity of Markov chains [21, 28, 29].”
Central limit theorem for Markov processes with spectral gap in the Wasserstein metric (2011, background): “The following theorem summarizes the results of [19, 20] that are of particular interest for us Theorem 6.”
An Introduction to 3D Stochastic Fluid Dynamics (2008, background): “In principle there could be others not in P, but this cannot happen in all those cases when uniqueness of the invariant measure is known (see [53] and the references therein).”
Exponential mixing for stochastic PDEs: the non-additive case (2005, background): “For instance, in [15], [27], the case of NS perturbed by a four dimensional noise is treated.”
On ergodicity of some Markov processes (2008, background): “A different approach to prove the uniqueness of an invariant measure for a stochastic evolution equation is based on the strong Feller property of the transition semigroup (see [4], [24], [7], and [12]) or in a more refined version on the asymptotic strong Feller property (see [13]).”
ANALYSIS OF SPDES ARISING IN PATH SAMPLING PART II: THE NONLINEAR CASE (2006, methodology): “The reason for introducing this process ρ is clear from the approximate integration by parts formula (see [12] for more details), which holds for every bounded”