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Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations
, 2006
"... 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 Ł ptype norm, but involves the derivative of the observable as ..."
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Cited by 21 (11 self)
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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 Ł ptype 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 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. 1
Harnack inequality and applications for stochastic evolution equations with monotone drifts
 J. Evol. Equat
"... As a Generalization to [37] where the dimensionfree Harnack inequality was established for stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations with general monotone drifts. Some ergodicity, compactness and contractivity proper ..."
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Cited by 5 (5 self)
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As a Generalization to [37] where the dimensionfree Harnack inequality was established for stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations with general monotone drifts. Some ergodicity, compactness and contractivity properties are established for the associated transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence of a spectral gap are also derived. As examples, the main results are applied to many concrete SPDEs such as stochastic reactiondiffusion equations, stochastic porous media equations and the stochastic pLaplace equation in Hilbert space.
Fine Properties of Stochastic Evolution Equations and Their Applications
, 2009
"... In this work, we aim to study some fine properties for a class of nonlinear SPDE within the variational framework. The results consist of three main parts. In the first part, we study the asymptotic behavior of nonlinear SPDE with small multiplicative noise. A FreidlinWentzell large deviation princ ..."
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In this work, we aim to study some fine properties for a class of nonlinear SPDE within the variational framework. The results consist of three main parts. In the first part, we study the asymptotic behavior of nonlinear SPDE with small multiplicative noise. A FreidlinWentzell large deviation principle is established for the distributions of solutions to a large class of SPDE, which include all stochastic evolution equations with monotone coefficients. In the second part, some properties of invariant measures and transition semigroups are investigated for SPDE with additive noise. The main tool is the dimensionfree Harnack inequality, which is established by using a coupling method and Girsanov transformation techniques. Subsequently, the Harnack inequality is used to derive the ergodicity, compactness and contractivity (e.g. hyperboundedness or ultraboundedness) for the associated transition semigroups. Moreover, the uniformly exponential convergence of the transition semigroup to the invariant measure and the existence of a spectral gap are also obtained. These results are first established for general stochastic evolution equations with strongly dissipative drift, e.g. stochastic reactiondiffusion equations, stochastic
Large deviations and Gallavotti–Cohen principle for dissipative PDE’s with rough noise
, 2014
"... We study a class of dissipative PDE’s perturbed by an unbounded kick force. Under some natural assumptions, the restrictions of solutions to integer times form a homogeneous Markov process. Assuming that the noise is rough with respect to the space variables and has a nondegenerate law, we prove t ..."
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We study a class of dissipative PDE’s perturbed by an unbounded kick force. Under some natural assumptions, the restrictions of solutions to integer times form a homogeneous Markov process. Assuming that the noise is rough with respect to the space variables and has a nondegenerate law, we prove that the system in question satisfies a large deviation principle (LDP) in τtopology. Under some additional hypotheses, we establish a Gallavotti–Cohen type symmetry for the rate function of an entropy production functional and the strict positivity and finiteness of the mean entropy production in the stationary regime. The latter result is applicable to PDE’s with strong nonlinear dissipation.
IRREDUCIBILITY AND UNIQUENESS OF STATIONARY DISTRIBUTION
, 902
"... Abstract. In this paper, we shall prove that the irreducibility in the sense of fine topology implies the uniqueness of invariant probability measures. It is also proven that this irreducibility is strictly weaker than the strong Feller property plus irreducibility in the sense of original topology, ..."
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Abstract. In this paper, we shall prove that the irreducibility in the sense of fine topology implies the uniqueness of invariant probability measures. It is also proven that this irreducibility is strictly weaker than the strong Feller property plus irreducibility in the sense of original topology, which is the usual uniqueness condition. 2000 MR subject classification: 60J45
Comparison of Tracking Algorithms for Single Layer Threshold Networks in the Presence of Random Drift
, 1995
"... . This paper analyzes the behavior of a variety of tracking algorithms for single layer threshold networks in the presence of random drift. We use a system identification model to model a target network where weights slowly change and a tracking network. Tracking algorithms are divided into conserva ..."
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. This paper analyzes the behavior of a variety of tracking algorithms for single layer threshold networks in the presence of random drift. We use a system identification model to model a target network where weights slowly change and a tracking network. Tracking algorithms are divided into conservative and nonconservative algorithms. For a random drift rate of fl, we find upper bounds for the generalization error of conservative algorithms that are O(fl 2=3 ) and for nonconservative algorithms that are O(fl). Bounds are found for the Perceptron tracker and the least mean square (LMS) tracker. Simulations show the validity of these bounds and also show that the bounds are tight when fl is small and the number of inputs n is large. These results show that the Perceptron tracker and the LMS tracker can work well in slowly changing nonstationary environments. 1 Random Drift Applied to Neural Networks 2 1 Introduction In this paper we study the performance of learning algorithms for s...
Comparison of Tracking Algorithms for Single Layer Threshold Networks in the Presence of Random Drift
"... Abstract — This paper analyzes the behavior of a variety of tracking algorithms for singlelayer threshold networks in the presence of random drift. We use a system identification model to model a target network where weights slowly change and a tracking network. Tracking algorithms are divided into ..."
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Abstract — This paper analyzes the behavior of a variety of tracking algorithms for singlelayer threshold networks in the presence of random drift. We use a system identification model to model a target network where weights slowly change and a tracking network. Tracking algorithms are divided into conservative and nonconservative algorithms. For a random drift rate of, we find upper bounds for the generalization error of conservative algorithms that are y @ PaQ A and for nonconservative algorithms that are y @ A. Bounds are found for the Perceptron tracker and the least mean square (LMS) tracker. Simulations show the validity of these bounds and show that the bounds are tight when is small and the number of inputs � is large. These results show that the Perceptron tracker and the LMS tracker can work well in slowly changing nonstationary environments. I.
Cylindrical
, 2009
"... Dispersion and collapse in stochastic velocity fields on a cylinder ..."
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