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15
Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré
 J. Func. Anal
, 1996
"... Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new ..."
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Cited by 25 (14 self)
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Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (LyapunovPoincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic FokkerPlanck equation recently studied by HérauNier, HelfferNier and Villani is in particular discussed in the final section.
Yet another look at Harris’ ergodic theorem for Markov chains
, 2008
"... C. Mattingly The aim of this note is to present an elementary proof of a variation of Harris’ ergodic theorem of Markov chains. This theorem, dating back to the fifties [Har56] essentially states that a Markov chain is uniquely ergodic if it admits a “small ” set (in a technical sense to be made pre ..."
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Cited by 11 (7 self)
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C. Mattingly The aim of this note is to present an elementary proof of a variation of Harris’ ergodic theorem of Markov chains. This theorem, dating back to the fifties [Har56] essentially states that a Markov chain is uniquely ergodic if it admits a “small ” set (in a technical sense to be made precise below) which is visited infinitely often. This gives an extension of the ideas of Doeblin to the unbounded state space setting. Often this is established by finding a Lyapunov function with “small ” level sets [Has80, MT93]. If the Lyapunov function is strong enough, one has a spectral gap in a weighted supremum norm [MT92, MT93]. In particular, its transition probabilities converge exponentially fast towards the unique invariant measure, and the constant in front of the exponential rate is controlled by the Lyapunov function [MT93]. Traditional proofs of this result rely on the decomposition of the Markov chain into excursions away from the small set and a careful analysis of the exponential tail of the length of these excursions [Num84, Cha89, MT92, MT93]. There have been other variations which have made use of Poisson equations or worked at getting explicit constants [KM05, DMR04, DMLM03]. The present proof is very direct, and relies instead on introducing a family of equivalent weighted norms indexed by a parameter β and to make an appropriate choice of this parameter that allows to combine in a very elementary way the two ingredients (existence of a Lyapunov function and irreducibility) that are crucial in obtaining a spectral gap. Use of a weighted totalvariation norm has been important since [MT92]. The original motivation of this proof was the authors ’ work on spectral gaps in Wasserstein metrics. The proof presented in this note is a version of our reasoning in the total variation setting which we used to guide the calculations in [HM08]. While we initially produced it for this purpose, we hope that it will be of interest in its own right. 1. Setting and main result Throughout this note, we fix a measurable space X and a Markov transition kernel P(x, ·) on X. We will use the notation P for the operators defined as usual on both the set of2 Martin Hairer and Jonathan Mattingly bounded measurable functions and the set of measures of finite mass by
A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs
"... We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with “polynomial ” nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander’s bracket condition holds at every point of this ..."
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Cited by 8 (7 self)
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We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with “polynomial ” nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander’s bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator Mt can be obtained. Informally, this bound can be read as “Fix any finitedimensional projection Π on a subspace of sufficiently regular functions. Then the eigenfunctions of Mt with small eigenvalues have only a very small component in the image of Π.” We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of “Wiener polynomials, ” where the coefficients are possibly nonadapted stochastic processes satisfying a Lipschitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris ’ lemma, which is unavailable in the present context. We conclude by showing that the twodimensional stochastic NavierStokes equations and a large class of reactiondiffusion equations fit the framework of our theory. Contents 1
An introduction to stochastic PDEs
 Lecture notes, 2009. URL http://arxiv.org/abs/0907.4178
"... ..."
2010: A simple framework to justify linear response theory
 Nonlinearity
"... The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced d ..."
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Cited by 5 (2 self)
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The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced dissipative stochastic dynamical systems is developed. The main results are formulated in an abstract setting and apply to suitable systems, in finite and infinite dimensions, that are of interest in climate change science and other applications. 1
Homogenization of periodic linear degenerate PDEs
, 2007
"... It is wellknown under the name of ‘periodic homogenization ’ that, under a centering condition of the drift, a periodic diffusion process on R d converges, under diffusive rescaling, to a ddimensional Brownian motion. Existing proofs of this result all rely on uniform ellipticity or hypoellipticit ..."
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Cited by 2 (1 self)
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It is wellknown under the name of ‘periodic homogenization ’ that, under a centering condition of the drift, a periodic diffusion process on R d converges, under diffusive rescaling, to a ddimensional Brownian motion. Existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion. In this paper, we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set. As a consequence, it is no longer the case that the effective diffusivity matrix is necessarily nondegenerate. It turns out that, provided that some very weak regularity conditions are met, the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion. In particular, this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank. We also discuss the application of our results to the homogenization of a class of elliptic and parabolic PDEs. 1
unknown title
, 2009
"... Asymptotic coupling and a weak form of Harris ’ theorem with applications to stochastic delay equations ..."
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Asymptotic coupling and a weak form of Harris ’ theorem with applications to stochastic delay equations
unknown title
, 2009
"... Asymptotic coupling and a weak form of Harris ’ theorem with applications to stochastic delay equations ..."
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Asymptotic coupling and a weak form of Harris ’ theorem with applications to stochastic delay equations
HEAT CONDUCTION NETWORKS: DISPOSITION OF HEAT BATHS AND INVARIANT MEASURE
, 902
"... Abstract. We consider a model of heat conduction networks consisting of oscillators in contact with heat baths at different temperatures. Our aim is to generalize the results concerning the existence and uniqueness of the stationnary state already obtained when the network is reduced to a chain of p ..."
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Abstract. We consider a model of heat conduction networks consisting of oscillators in contact with heat baths at different temperatures. Our aim is to generalize the results concerning the existence and uniqueness of the stationnary state already obtained when the network is reduced to a chain of particles. Using Lasalle’s principle, we establish a condition on the disposition of the heat baths among the network that ensures the uniqueness of the invariant measure. We will show that this condition is sharp when the oscillators are linear. Moreover, when the interaction between the particles is stronger than the pinning, we prove that this condition implies the existence of the invariant measure. 1. Definitions and Results 1.1. The motivations. We consider an arbitrary graph. At each vertex of this graph, there is a particle interacting with the substrate and with its neighbours. Among these particles, some are linked to heat baths; an OrnsteinUhlenbeck process models this interaction. Given this graph, we establish conditions on the disposition of the heat baths that entails existence and uniqueness of the invariant measure. When the graph is reduced to a chain, each extremal particle is connected to a heat bath.
AN EFFICIENT SECOND ORDER IN TIME SCHEME FOR APPROXIMATING LONG TIME STATISTICAL PROPERTIES OF THE TWO DIMENSIONAL NAVIERSTOKES EQUATIONS
"... Abstract. We investigate the long time behavior of the following efficient second order in time scheme for the 2D NavierStokes equation in a periodic box: 3ωn+1 − 4ωn + ωn−1 + ∇ 2k ⊥ (2ψ n − ψ n−1) · ∇(2ω n − ω n−1) − ν∆ω n+1 = f n+1, −∆ψ n = ω n. The scheme is a combination of a 2nd order in tim ..."
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Abstract. We investigate the long time behavior of the following efficient second order in time scheme for the 2D NavierStokes equation in a periodic box: 3ωn+1 − 4ωn + ωn−1 + ∇ 2k ⊥ (2ψ n − ψ n−1) · ∇(2ω n − ω n−1) − ν∆ω n+1 = f n+1, −∆ψ n = ω n. The scheme is a combination of a 2nd order in time backwarddifferentiation (BDF) and a special explicit AdamsBashforth treatment of the advection term. Therefore only a linear constant coefficient Poisson type problem needs to be solved at each time step. We prove uniform in time bounds on this scheme in ˙ L2, H ˙ 1 per and ˙ H2 per provided that the timestep is sufficiently small. These time uniform estimates further lead to the convergence of long time statistics (stationary statistical properties) of the scheme to that of the NSE itself at vanishing timestep. Fully discrete schemes with either Galerkin Fourier or collocation Fourier spectral method are also discussed. Key words. NavierStokes equations, invariant measures, long time statistical properties, second order in time scheme, long time global stability AMS subject classifications. 1. Introduction. It