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Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations
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Hairer, Martin and Mattingly, Jonathan Christopher (2008) Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Annals of Probability, Volume 36 (Number 6). pp. 2050-2091. doi:10.1214/08-AOP392 ISSN 0091-1798.
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Official URL: http://dx.doi.org/10.1214/08-AOP392
Abstract
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.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Navier-Stokes equations, Stochastic partial differential equations, Ergodic theory | ||||
Journal or Publication Title: | Annals of Probability | ||||
Publisher: | Institute of Mathematical Statistics | ||||
ISSN: | 0091-1798 | ||||
Official Date: | November 2008 | ||||
Dates: |
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Volume: | Volume 36 | ||||
Number: | Number 6 | ||||
Number of Pages: | 42 | ||||
Page Range: | pp. 2050-2091 | ||||
DOI: | 10.1214/08-AOP392 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC), National Science Foundation (U.S.) (NSF), Alfred P. Sloan Foundation | ||||
Grant number: | EP/D071593/1 (EPSRC), DMS-04-49910 (NSF) |
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