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91
Uniqueness of the Invariant Measure for a Stochastic PDE Driven by Degenerate Noise
, 2001
"... We consider the stochastic GinzburgLandau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The lowlying frequencies are then only connected to this forcing through the nonlinear (cubic) term of the GinzburgLandau equation. Under these assumpti ..."
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Cited by 40 (11 self)
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We consider the stochastic GinzburgLandau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The lowlying frequencies are then only connected to this forcing through the nonlinear (cubic) term of the GinzburgLandau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof combine a controllability argument for the lowlying frequencies with an infinite dimensional version of the Malliavin calculus to show positivity and regularity of the invariant measure. This then implies the uniqueness of that measure. Contents 1 Introduction 2 2 Some Preliminaries on the Dynamics 5 3 Controllability 6 4 Strong Feller Property and Proof of Theorem 1.1 9 5 Regularity of the Cutoff Process 11 5.1 Splitting and Interpolation Spaces . . . . . . . . . . . . . . . . . . . 12 5.2 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.3 Smoothing Properties of the...
Exponential Mixing Properties of Stochastic PDEs Through Asymptotic Coupling
 Probab. Theory Related Fields
, 2001
"... We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions amount essentially to the fact that the equation transmits ..."
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Cited by 25 (7 self)
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We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions amount essentially to the fact that the equation transmits the noise to all its determining modes. Several examples are investigated, including some where the noise does not act on every determining mode directly.
The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Part 1: The Stochastic semiflow, Part 2: Existence of stable and unstable manifolds
 98, Memoirs of the American Mathematical Society
, 2002
"... Abstract. The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the longterm behav ..."
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Cited by 22 (12 self)
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Abstract. The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the longterm behavior of the solution field near stationary points. The analysis falls in two parts 1, 2. In Part 1, we prove general existence and compactness theorems for C kcocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with Lipschitz and nonLipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinitedimensional noise. In Part 2, stationary solutions are viewed as cocycleinvariant random points in the infinitedimensional state space. The pathwise local structure of solutions of semilinear see’s and spde’s near stationary solutions is described in terms of the almost sure longtime behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see’s and spde’s (Theorems 2.4.12.4.4). These results give smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution of the underlying stochastic equation. The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde. Furthermore, the local stable and unstable manifolds intersect transversally at the stationary point, and the unstable manifolds have fixed finite dimension. The proof uses infinitedimensional multiplicative ergodic theory techniques, interpolation and perfection arguments (Theorem 2.2.1).
From Metropolis to Diffusions: Gibbs States and Optimal Scaling. Stochastic Process
 Appl
, 2000
"... Abstract. This paper investigates the behaviour of the random walk Metropolis algorithm in high dimensional problems. Here we concentrate on the case where the components in the target density is a spatially homogeneous Gibbs distribution with finite range. The performance of the algorithm is strong ..."
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Cited by 19 (10 self)
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Abstract. This paper investigates the behaviour of the random walk Metropolis algorithm in high dimensional problems. Here we concentrate on the case where the components in the target density is a spatially homogeneous Gibbs distribution with finite range. The performance of the algorithm is strongly linked to the presence or absence of phase transition for the Gibbs distribution; the convergence time being approximately linear in dimension for problems where phase transition is not present. Related to this, there is an optimal way to scale the variance of the proposal distribution in order to maximise the speed of convergence of the algorithm. This turns out to involve scaling the variance of the proposal as the reciprocal of dimension (at least in the phase transition free case). Moreover the actual optimal scaling can be characterised in terms of the overall acceptance rate of the algorithm, the maximising value being 0.234, the value as predicted by studies on simpler classes of target density. The results are proved in the framework of a weak convergence result, which shows that the algorithm actually behaves like an infinite dimensional diffusion process in high dimensions.
Markov solutions for the 3D stochastic NavierStokes equations with state dependent noise, available on the arXiv preprint archive at the web address http://www.arxiv.org/abs/math.AP/0512361
"... Abstract: We construct a Markov family of solutions for the 3D NavierStokes equation perturbed by a non degenerate noise. We improve the result of [3] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a ..."
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Cited by 16 (5 self)
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Abstract: We construct a Markov family of solutions for the 3D NavierStokes equation perturbed by a non degenerate noise. We improve the result of [3] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a state dependant noise. Another feature of this work is that we greatly simplify the proofs of [3].
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 15 (7 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
Existence and stability for FokkerPlanck equations with logconcave reference measure
, 2007
"... We study Markov processes associated with stochastic differential equations, whose nonlinearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability pr ..."
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Cited by 14 (5 self)
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We study Markov processes associated with stochastic differential equations, whose nonlinearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a FokkerPlanck equation as the steepest descent flow of the relative Entropy functional in the space of probability measures, endowed with the Wasserstein distance. Applications include stochastic partial differential equations and convergence of equilibrium fluctuations for a class of random interfaces.
Dissipative Quasigeostrophic Dynamics under Random Forcing
, 1998
"... The quasigeostrophic model is a simplified geophysical fluid model at asymptotically high rotation rate or at small Rossby number. We consider the quasigeostrophic equation with dissipation under random forcing in bounded domains. We show that global unique solutions exist for appropriate initial d ..."
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Cited by 12 (12 self)
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The quasigeostrophic model is a simplified geophysical fluid model at asymptotically high rotation rate or at small Rossby number. We consider the quasigeostrophic equation with dissipation under random forcing in bounded domains. We show that global unique solutions exist for appropriate initial data. Unlike the deterministic quasigeostrophic equation whose wellposedness is wellknown, there seems no rigorous result on global existence and uniqueness of the randomly forced quasigeostrophic equation. Our work provides such a rigorous result on global existence and uniqueness, under very mild conditions.
Nonlinear feedback systems perturbed by noise: steadystate probability distributions and optimal control
 IEEE Trans. Automat. Control
, 2000
"... Abstract—In this paper we describe a class of nonlinear feedback systems perturbed by white noise for which explicit formulas for steadystate probability densities can be found. We show that this class includes what has been called monotemperaturic systems in earlier work and establish relationship ..."
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Cited by 11 (5 self)
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Abstract—In this paper we describe a class of nonlinear feedback systems perturbed by white noise for which explicit formulas for steadystate probability densities can be found. We show that this class includes what has been called monotemperaturic systems in earlier work and establish relationships with Lyapunov functions for the corresponding deterministic systems. We also treat a number of stochastic optimal control problems in the case of quantized feedback, with performance criteria formulated in terms of the steadystate probability density. Index Terms—Nonlinear feedback system, quantizer, steadystate probability density, stochastic optimal control, white noise. I.
STOCHASTIC DYNAMICS OF A COUPLED ATMOSPHERE–OCEAN MODEL
"... Abstract. The investigation of the coupled atmosphereocean system is not only scientifically challenging but also practically important. We consider a coupled atmosphereocean model, which involves hydrodynamics, thermodynamics, and random atmospheric dynamics due to short time influences at the ai ..."
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Cited by 10 (9 self)
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Abstract. The investigation of the coupled atmosphereocean system is not only scientifically challenging but also practically important. We consider a coupled atmosphereocean model, which involves hydrodynamics, thermodynamics, and random atmospheric dynamics due to short time influences at the airsea interface. We reformulate this model as a random dynamical system. First, we have shown that the asymptotic dynamics of the coupled atmosphereocean model is described by a random climatic attractor. Second, we have estimated the atmospheric temperature evolution under oceanic feedback, in terms of the freshwater flux, heat flux and the external fluctuation at the airsea interface, as well as the earth’s longwave radiation coefficient and the shortwave solar radiation profile. Third, we have demonstrated that this system has finite degree of freedom by presenting a finite set of determining functionals in probability. Finally, we have proved that the coupled atmosphereocean model is ergodic under suitable conditions for physical parameters and randomness, and thus for any observable of the coupled