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Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations. (English) Zbl 1173.37005
The authors develop a general method to prove the existence of spectral gaps for Markov semigroups on Banach spaces. A key point is the norm, in which convergence to the invariant measure is established. It involves the derivative of the observable as well and hence can be seen as a type of 1-Wasserstein distance. To be more precise, the observables are measured by a weighted $$W^{1,\infty}$$-norm. This turns out to be a suitable approach for infinite-dimensional spaces where the usual Harris or Döblin conditions, which are geared toward total variation convergence, often fail to hold.
The results are based on “gradient estimates”, i.e., bounds on the derivative of the semigroup, that involve derivatives of the observables. Bounds of this type were established by M. Hairer and J. C. Mattingly [Ann. Math. (2) 164, No. 3, 993–1032 (2006; Zbl 1130.37038)], in order to introduce the asymptotic strong Feller property.
In the first part of this paper semigroups are considered with a uniform behavior which one can view as the analog of Döblin’s condition. In the second part the behavior is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition.
The general results are applied to the two-dimensional stochastic Navier-Stokes equations in vorticity form, even in situations where the forcing is extremely degenerate. It is shown that the invariant measure depends continuously on the viscosity and the structure of the forcing.

##### MSC:
 37A30 Ergodic theorems, spectral theory, Markov operators 37A25 Ergodicity, mixing, rates of mixing 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 76D05 Navier-Stokes equations for incompressible viscous fluids 37H10 Generation, random and stochastic difference and differential equations 35R60 PDEs with randomness, stochastic partial differential equations
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