SPECTRAL GAPS IN WASSERSTEIN DISTANCES AND THE 2D STOCHASTIC NAVIER-STOKES EQUATIONS

• 成果类型: Article
• 署名作者: Hairer, Martin; Mattingly, Jonathan C.
• 署名单位: University of Warwick; Duke University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP392
• 发表日期: 2008
• 页码: 2050-2091
• 关键词: infinite-dimensional systems; malliavin calculus; coupling approach; degenerate noise; ergodicity; DYNAMICS; pdes; approximation; burgers

摘要

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.