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19
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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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.
ANISOTROPIC HYPOELLIPTIC ESTIMATES FOR LANDAUTYPE OPERATORS
"... Abstract. We establish global hypoelliptic estimates for linear Landautype operators. Linear Landautype equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By intr ..."
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Abstract. We establish global hypoelliptic estimates for linear Landautype operators. Linear Landautype equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to various hypoelliptic kinetic equations, we establish for linear Landautype operators optimal global hypoelliptic estimates with loss of 4/3 derivatives in a Sobolev scale which is exactly related to the anisotropy of the diffusion. 1.
Uniform bounds and exponential time decay . . .
, 2005
"... We consider the nonlinear 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 in weighted spaces and for small charge we find an explicit ..."
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We consider the nonlinear 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 in weighted spaces and for small charge we find an explicit exponential rate of convergence to the equilibrium in terms of the Witten Laplacian associated to the linear equation.
Exponential return to equilibrium for hypoelliptic quadratic systems
 J. Funct. Anal
"... Abstract. We study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators. Quadratic operators are nonselfadjoint differential operators with complexvalued quadratic symbols. Under appropriate assumptions, a complete description of the spectrum o ..."
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Cited by 1 (1 self)
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Abstract. We study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators. Quadratic operators are nonselfadjoint differential operators with complexvalued quadratic symbols. Under appropriate assumptions, a complete description of the spectrum of such operators is given and the exponential return to equilibrium with sharp estimates on the rate of convergence is proven. Some applications to the study of chains of oscillators and the generalized Langevin equation are given. 1.
Homogenization for Inertial Particles in a Random Flow
, 2008
"... We study the problem of homogenization for inertial particles moving in a time dependent random velocity field and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large–scale, long–time behavior of the inertial particles is governed by an effect ..."
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We study the problem of homogenization for inertial particles moving in a time dependent random velocity field and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large–scale, long–time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone. This is achieved by the use of a formal multiple scales expansion in the scale parameter. The expansion relies on the hypoellipticity of the underlying diffusion. An expression for the diffusivity tensor is found and various of its properties are studied. The results of the formal multiscale analysis are justified rigorously by the use of the martingale central limit theorem. Our theoretical findings are supported by numerical investigations where we study the parametric dependence of the effective diffusivity on the various non–dimensional parameters of the problem. 1
unknown title
, 2003
"... Low regularity solutions to a gently stochastic nonlinear wave equation in nonequilibrium statistical mechanics ..."
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Low regularity solutions to a gently stochastic nonlinear wave equation in nonequilibrium statistical mechanics
SECOND QUANTIZATION AND THE L pSPECTRUM OF NONSYMMETRIC ORNSTEINUHLENBECK OPERATORS
, 2005
"... Abstract. The spectra of the second quantization and the symmetric second quantization of a strict Hilbert space contraction are computed explicitly and shown to coincide. As an application, we compute the spectrum of the nonsymmetric OrnsteinUhlenbeck operator L associated with the infinitedimens ..."
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Abstract. The spectra of the second quantization and the symmetric second quantization of a strict Hilbert space contraction are computed explicitly and shown to coincide. As an application, we compute the spectrum of the nonsymmetric OrnsteinUhlenbeck operator L associated with the infinitedimensional Langevin equation dU(t) = AU(t) dt + dW(t) where A is the generator of a strongly continuous semigroup on a Banach space E and W is a cylindrical Wiener process in E. Assuming the existence of an invariant measure µ for L, under suitable assumptions on A we show that the spectrum of L in the space L p (E, µ) (1 < p < ∞) is given by { ∑n σ(L) = j=1 kjzj: kj ∈ N, zj ∈ σ(Aµ); j = 1,..., n; n ≥ 1, where Aµ is the generator of a Hilbert space contraction semigroup canonically associated with A and µ. We prove that the assumptions on A are always satisfied in the strong Feller case and in the finitedimensional case. In the latter case we recover the recent MetafunePallaraPriola formula for σ(L). 1.