We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M.

As part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations, begun in [13], we derive estimates on the rate of convergence to equilibrium for solutions of the Boltzmann equation, like O(t-∞). Our results hold...

Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications. 1.

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiz...

We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probablity measures which satisfy these inequalities.

We derive concentration inequalities for functions of the empirical measure of eigenvalues for large, random, self adjoint matrices, with not necessarily Gaussian entries. The results presented apply in particular to non-Gaussian Wigner and Wishart matrices. We also provide concentration bounds for non commutative functionals of random matrices. 1 Introduction and statement of results Consider a random N N Hermitian matrix X with i.i.d. complex entries (except for the symmetry constraint) satisfying a moment condition. It is well known since Wigner [28] that the spectral measure of N 1=2 X converges to the semicircle law. This observation has been generalized to a large class of matrices, e.g. sample covariance matrices of the form XRX where R is a deterministic diagonal matrix ([19]), band matrices (see [5, 16, 20]), etc. For the Wigner case, this convergence has been supplemented by Central Limit Theorems, see [15] for the case of Gaussian entries and [17], [22] for the gen...

this paper, our local specification will come from a shift invariant, finite range Gibbs potential \Phi j f\Phi X gX2F . That is, (1) for each X 2 F, \Phi X 2 CX(\Omega\Gamma2

An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite-dimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even non-convexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow. 1

In this paper, we find optimal constants of a special class of Gagliardo-Nirenberg type inequalities which turns out to interpolate between the classical Sobolev inequality and the Gross logarithmic Sobolev inequality. These inequalities provide an optimal decay rate (measured by entropy methods) of the intermediate asymptotics of solutions to nonlinear diffusion equations.

. We prove that for finite range discrete spin systems on the two dimensional lattice Z 2 , the (weak) mixing condition which follows, for instance, from the DobrushinShlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the Dobrushin-Shlosman complete analiticity condition, but restricted to all squares in the lattice, or, more generally, to all sets multiple of a large enough square. The key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk, because of the weak mixing condition, or along the boundary because it is one dimensional. As a consequence we obtain for ferromagnetic Ising-type systems proofs that several nice properties hold arbitrarily close to the critical temperature; these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic Sobolev constant and rapid convergence to equilibrium of the assoc...