Browsing by Subject "ERGODICITY"

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Open Access
Spectral gaps in wasserstein distances and the 2d stochastic navier-stokes equations
(Annals of Probability, 2008-11-01) Hairer, Martin; Mattingly, Jonathan C
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 Ł p-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 as 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 degenerate. 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. © Institute of Mathematical Statistics, 2008.
Open Access
Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations
(Communications in Contemporary Mathematics, 2005-10-01) Bakhtin, Y; Mattingly, JC
We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier-Stokes equation and stochastic Ginsburg-Landau equation. © World Scientific Publishing Company.