Math @ Duke

Publications [#243868] of Jonathan C. Mattingly
search arxiv.org.Papers Published
 with Martin Hairer,, Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations,
Annals of Probability, vol. 36 no. 6
(2008),
pp. 9931032, Institute of Mathematical Statistics [MR2478676], [math.PR/0602479], [doi]
(last updated on 2019/06/16)
Abstract: We develop a general method that allows to
show 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 L^ptype norm,
but involves the derivative of the observable
as well and hence can be seen as a type of
1Wasserstein distance. This turns out to be
a suitable approach for infinitedimensional
spaces where the usual Harris or Doeblin
conditions, which are geared to total
variation convergence, regularly fail to
hold. In the first part of this paper, we
consider semigroups that have uniform
behaviour which one can view as an extension
of Doeblin's condition. We then proceed to
study situations where the behaviour 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
twodimensional stochastic NavierStokers
equations, even in situations where the
forcing is extremely degenerate. Using the
convergence result, we show shat the
stochastic NavierStokes equations' invariant
measures depend continuously on the viscosity
and the structure of the forcing.


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