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42
Isotropic hypoellipticity and trend to the equilibrium for the FokkerPlanck equation with high degree potential
, 2002
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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 ..."
Abstract

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 Convergence to NonEquilibrium Stationary States in Classical Statistical Mechanics
 Comm. Math. Phys
, 2001
"... We continue the study of a model for heat conduction [6] consisting of a chain of nonlinear oscillators coupled to two Hamiltonian heat reservoirs at dierent temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the d ..."
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Cited by 33 (6 self)
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We continue the study of a model for heat conduction [6] consisting of a chain of nonlinear oscillators coupled to two Hamiltonian heat reservoirs at dierent temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the innite dimensional dynamics to a nitedimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators. 1 Introduction In its present state, nonequilibrium statistical mechanics is lacking the rm theoretical foundations that equilibrium statistical mechanics has. This is due, perhaps, to the extremely great variety of physical phenomena that nonequilibrium statistical mechanics describes. We will concentrate here on a system which is maintained, by suitable forces, in a state far from equilibrium. In such an idealization, the nonequilibrium phenome...
Fluctuations of the entropy production in anharmonic chains
 Ann. Henri Poincare
, 2002
"... Abstract. We prove the GallavottiCohen fluctuation theorem for a model of heat conduction through a chain of anharmonic oscillators coupled to two Hamiltonian reservoirs at different temperatures. 1 ..."
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Cited by 22 (4 self)
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Abstract. We prove the GallavottiCohen fluctuation theorem for a model of heat conduction through a chain of anharmonic oscillators coupled to two Hamiltonian reservoirs at different temperatures. 1
Asymptotic Behavior of Thermal Nonequilibrium Steady States for a Driven Chain of Anharmonic Oscillators
 COMMUN. MATH. PHYS.
, 2000
"... We consider a model of heat conduction introduced in [6], which consists of a finite nonlinear chain coupled to two heat reservoirs at different temperatures. We study the low temperature asymptotic behavior of the invariant measure. We show that, in this limit, the invariant measure is characteriz ..."
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Cited by 21 (4 self)
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We consider a model of heat conduction introduced in [6], which consists of a finite nonlinear chain coupled to two heat reservoirs at different temperatures. We study the low temperature asymptotic behavior of the invariant measure. We show that, in this limit, the invariant measure is characterized by a variational principle. The main technical ingredients are some control theoretic arguments to extend the Freidlin–Wentzell theory of large deviations to a class of degenerate diffusions.
Spectral Properties of Hypoelliptic Operators
 Commun. Math. Phys
, 2003
"... We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = i X i + X0 + f , where the X j denote first order differential operators, f is a function with at most polynomial growth, and X i denotes the formal adjoint of X i in L . For any # > 0 we show t ..."
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Cited by 18 (0 self)
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We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = i X i + X0 + f , where the X j denote first order differential operators, f is a function with at most polynomial growth, and X i denotes the formal adjoint of X i in L . For any # > 0 we show that an inequality of the form C(#u#0,# + + iy)u#0,0 ) holds for suitable # and C which are independent of R, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the FokkerPlanck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of H erau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+ iy # y c, # (0, 1], c R}.
Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats.
, 802
"... We discuss the DonskerVaradhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the DonskerVaradhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, ..."
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Cited by 6 (0 self)
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We discuss the DonskerVaradhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the DonskerVaradhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary state as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic coupling. 1 Introduction. Attempts to study large systems out of equilibrium through fluctuation theory has received a lot of attention in recent years [2, 3, 4, 5, 16, 18, 19, 20]. In a recent series of papers [27, 8, 28, 29, 30], it has been understood that in random systems driven out of equilibrium, the theory of large deviations provides naturally a variational
Statistical Mechanics of anharmonic lattices
 In Advances in Differential Equations and Mathematical Physics, Contemporary Mathematics 327
, 2003
"... Abstract. We discuss various aspects of a series of recent works on the nonequilibrium stationary states of anharmonic crystals coupled to heat reservoirs (see also [7]). We expose some of the main ideas and techniques and also present some open problems. 1. ..."
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Cited by 5 (1 self)
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Abstract. We discuss various aspects of a series of recent works on the nonequilibrium stationary states of anharmonic crystals coupled to heat reservoirs (see also [7]). We expose some of the main ideas and techniques and also present some open problems. 1.
Ergodicity for langevin processes with degenerate diffusion in momentums. Submitted; arXiv
, 2007
"... This paper presents sufficient conditions for proving ergodicity of noisedriven dynamical systems. The essential conditions are weak irreducibility and closure under second randomization of the driving noise. With these conditions one can ascertain ergodicity of Langevin processes even if the diffu ..."
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Cited by 4 (1 self)
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This paper presents sufficient conditions for proving ergodicity of noisedriven dynamical systems. The essential conditions are weak irreducibility and closure under second randomization of the driving noise. With these conditions one can ascertain ergodicity of Langevin processes even if the diffusion and drift matrices associated to the momentums are degenerate. The paper illustrates how to check these conditions practically in the context of a simple mechanical system governed by Langevin equations (a simple stochastic rigid body system). 1