Abstract. We consider a stochastic Klein-Gordon wave equation modeling heat flow in a linear field that is coupled to thermal reservoirs at different temperatures. We discuss, in a perturbative context, the approach to a stationary, non-equilibrium state, and show that the state is supported on field configurations which are Hölder continuous, with any exponent less than 1/2. We determine the heat flux to lowest order in perturbation theory. 1.

The mathematical physics of mechanical systems in thermal equilibrium is a well studied, and relatively easy, subject, because the Gibbs distribution is in general an adequate guess for the equilibrium state. On the other hand, the mathematical physics of non-equilibrium systems, such as that of a chain of masses connected with springs to two (infinite) heat reservoirs is more difficult, precisely because no such a priori guess exists. Recent work has, however, revealed that under quite general conditions, such states can not only be shown to exist, but are unique, using the H"ormander conditions and controllability. Furthermore, interesting properties, such as energy flux, exponentially fast convergence to the unique state, and fluctuations of that state have been successfully studied. Finally, the ideas used in these studies can be extended to certain stochastic PDE's using Malliavin calculus to prove regularity of the process.

Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential

Abstract not found

We consider the stochastic Ginzburg-Landau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The low-lying frequencies are then only connected to this forcing through the non-linear (cubic) term of the Ginzburg-Landau 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 low-lying 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.

Abstract: We consider the non-linear VPFP system with a coulombian repulsive interaction potential and a generic confining potential in space dimension d ≥ 3. Using spectral and kinetic methods we prove the existence and uniqueness of a mild solution with bounds uniform in time in weighted spaces, and for small total charge we find an explicit exponential rate of convergence toward the equilibrium in terms of the Witten Laplacian associated to the linear equation. Résumé: On considère le système de Vlasov-Poisson-Fokker-Planck avec un potentiel Coulombien répulsif et un potentiel confinant générique en dimension d ≥ 3. Avec des méthodes spectrales et cinétiques on prouve l’existence et l’unicité d’une solution douce dans des espaces à poids, bornée uniformément en temps, et pour petite charge totale on trouve un taux de retour exponentiel explicite vers l’équilibre en fonction du Laplacien de Witten associé à l’équation linéaire.

and long time behavior of the Fokker-Planck equation

Abstract. In this note we consider a chain of N oscillators, whose ends are in contact with two heat baths at different temperatures. Our main result is the exponential convergence to the unique invariant probability measure (the stationary state). We use the Lyapunov’s function technique of Rey-Bellet and coauthors [11, 8, 13, 12, 4, 5], with different model of heat baths, and adapt these techniques to two new case recently considered in the literature by Bernardin and Olla [2] and Lefevere and

We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for ‘extreme ’ non-equilibrium statistical mechanics. We provide a full picture of the long-time behaviour of such a system, including the existence / non-existence of a non-equilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state. Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is ‘too stiff’, then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds.

Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications