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.

this paper is to provide such general-purpose inequalities. Our approach is based on a generalization of Ledoux's entropy method (see [26, 28]). Ledoux's method relies on abstract functional inequalities known as logarithmic Sobolev inequalities and provide a powerful tool for deriving exponential inequalities for functions of independent random variables, see Boucheron, Massart, and AMS 1991 subject classifications. Primary 60E15, 60C05, 28A35; Secondary 05C80 Key words and phrases. Moment inequalities, Concentration inequalities; Empirical processes; Random graphs Supported by EU Working Group RAND-APX, binational PROCOPE Grant 05923XL The work of the third author was supported by the Spanish Ministry of Science and Technology and FEDER, grant BMF2003-03324 Lugosi [6, 7], Bousquet [8], Devroye [14], Massart [30, 31], Rio [36] for various applications. To derive moment inequalities for general functions of independent random variables, we elaborate on the pioneering work of Latala and Oleszkiewicz [25] and describe so-called #-Sobolev inequalities which interpolate between Poincare's inequality and logarithmic Sobolev inequalities (see also Beckner [4] and Bobkov's arguments in [26])

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Abstract. Concentration inequalities deal with deviations of functions of independent random variables from their expectation. In the last decade new tools have been introduced making it possible to establish simple and powerful inequalities. These inequalities are at the heart of the mathematical analysis of various problems in machine learning and made it possible to derive new efficient algorithms. This text attempts to summarize some of the basic tools. 1

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Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1.

The flashing rachet is the simplest example of diffusion mediated transport as well as the suggested mechanism for a class of protein motors. Here we briefly explain these concepts and give an entropy based argument for existence and uniqueness of a model problem. We also examine the features of the system that lead to transport 1

This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy–entropy production relation improves the known inequalities of this type. The corresponding generalized Poincaré type inequalities with weights are derived. Best constants are compared to the usual Poincaré constant. Key words and phrases: Sobolev inequality – Poincaré inequality – entropy method

This paper is concerned with entropy methods for linear drift-diffusion equations with explicitly time-dependent or degenerate coefficients. Our goal is to establish a list of various qualitative properties of the solutions. The motivation for this study comes from a model for molecular motors, the so-called Brownian ratchet, and from a nonlinear equation arising in traffic flow models, for which complex long time dynamics occurs. General results are out of the scope of this paper, but we deal with several examples corresponding to most of the expected behaviors of the solutions. We first prove a contraction property for general entropies which is a useful tool for uniqueness and for the convergence to some eventually time-dependent large time asymptotic solutions. Then we focus on power law and logarithmic relative entropies. When the diffusion term is of the type ∇(|x | α ∇·), we prove that the inequality relating the entropy with the entropy production term is a Hardy-Poincaré type inequality, that we establish. Here we assume that α ∈ (0, 2] and the limit case α = 2 appears as a threshold for the method. As a consequence, we obtain an exponential decay of the relative entropies. In the case of time-periodic coefficients, we prove the existence of a unique time-periodic solution which attracts all other solutions. The case of a degenerate diffusion coefficient taking the form |x | α with α> 2 is also studied. The Gibbs state exhibits a non integrable singularity. In this case concentration phenomena may occur, but we conjecture that an additional time-dependence restores the smoothness of the asymptotic solution.

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