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We develop a general method to prove the existence of spectral gaps forMarkov semigroups on Banach spaces. Unlike most previous work, the type of normwe consider for this analysis is neither a weighted supremum norm nor an${\L}^p$-type norm, but involves the derivative of the observable as well andhence can be seen as a type of 1-Wasserstein distance. This turns out to be asuitable approach for infinite-dimensional spaces where the usual Harris orDoeblin conditions, which are geared toward total variation convergence, oftenfail to hold. In the first part of this paper, we consider semigroups that haveuniform behavior which one can view as the analog of Doeblin's condition. Wethen proceed to study situations where the behavior is not so uniform, but thesystem has a suitable Lyapunov structure, leading to a type of Harriscondition. We finally show that the latter condition is satisfied by thetwo-dimensional stochastic Navier--Stokes equations, even in situations wherethe forcing is extremely degenerate. Using the convergence result, we show thatthe stochastic Navier--Stokes equations' invariant measures depend continuouslyon the viscosity and the structure of the forcing.

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