Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations

By Martin Hairer, Jonathan C. Mattingly. 2006

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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 LP-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 its 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 de generate. 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.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/41c02f010f3d9dde559904a2ccc0fcbffdf2add2

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Citations in the corpus

2D Stochastic Navier–Stokes Equations with a Time-Periodic Forcing Term (2008, background): “Recently much progress has been obtained in the study of existence, uniqueness and ergodicity of the associated invariant measures ([3], [15], [17], [18], [19], [20], [21], [11], [22], [23]) when f(t) = 0.”
Exponential ergodicity for Markov processes with random switching (2013, methodology): “ Theorem which yields geometric ergodicity under the existence of a Lyapunov function and a modiﬁed “small set” condition. This theorem was previously applied to the stochastic Navier–Stokes equation [22], stochastic delay diﬀerential equations [24], and linear response theory [21]. It is an extensi”
A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs (2008, methodology): “[ HM08 , Theorem 2.5] yields the following corollary to Theorem 8.17.”
Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré (2007, background): “In the same spirit some result for some Wasserstein distance is obtained in [16].”
Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes (2016, background): “A breakthrough toward the applicability to high- and infinite-dimensional models has been made by Hairer and Mattingly in [29], and in the subsequent publications [25, 30].”
Central limit theorem for Markov processes with spectral gap in the Wasserstein metric (2011, methodology): “In that case one can use the regularization result of [20], see Proposition 5.”
A simple framework to justify linear response theory (2009, background): “ a space of 1 2 -Ho¨lder continuous functions, which is useless for our purpose. On the other hand,a spectral gapin a space of the type (2.1) was obtainedfor the stochastic Navier-Stokes equations in [HM08]. Inthissection,weobtainaslightlydifferentsetofconditionsunderwhichaspectral gap in a norm of ”
Structural properties of semilinear SPDEs driven by cylindrical stable processes (2008, methodology): “eller property holds in rather special situation. It seems thus of great interest to develop the concept of asymptotic strong Feller property in the case of SPDEs with L´evy noise (compare with [12], [13] and [14]). Theorem 5.9. Assume that Hypotheses 4.2, 5.1 and 5.6 hold. Then, for any t>0, the”
Elliptic and parabolic equations for measures (2009, background): “, the book [67]), especially connected with stochastic partial differential equations (see [11], [52], [57], [58], [62], [63], [70], [74], [79], [82], [90], [91], [128], [150], [163]), infinite gradient systems, Gibbs measures, and stochastic quantization (see [5], [6], [56], [83], [85], [86], [96],”
Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction (2013, background): “Later, Hairer and Mattingly have proved that under the same assumptions a spectral gap holds and exponential convergence to equilibrium is thus also true for very degenerate noises ([39]).”
Generalized couplings and ergodic rates for SPDEs and other Markov models (2018, methodology): “Note that exponential ergodic bounds had been obtained previously for the Navier-Stokes equation only on a 2-dimensional torus, see [31] and [21].”
Quantitative contraction rates for Markov chains on general state spaces (2018, background): “loped an analogue to Harris’ Theorem assuming only local strict contractivity in a Kantorovich distance on the “small” set and a Lyapunov condition combined with non-strict contractivity outside, cf. [22, 24]. Meanwhile there have been numerous extensions and applications of their result [25, 6, 3, ”
On ergodicity of some Markov processes (2008, methodology): “r a stochastic evolution equation is based on the strong Feller property of the transition semigroup (see [3, 6, 10] and [22]) or, in a more reﬁned form, on the asymptotic strong Feller property (see [11, 12, 19]). In our Theorem 3, we do not require either of these properties of the corresponding s”