Results 1  10
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11
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré
 J. Func. Anal
, 1996
"... Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new ..."
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Cited by 30 (16 self)
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Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (LyapunovPoincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic FokkerPlanck equation recently studied by HérauNier, HelfferNier and Villani is in particular discussed in the final section.
Hypercontractivity for perturbed diffusion semigroups
 ANN. FAC. DES SC. DE TOULOUSE
, 2005
"... µ being a nonnegative measure satisfying some LogSobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some LogSobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary conditio ..."
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Cited by 22 (15 self)
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µ being a nonnegative measure satisfying some LogSobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some LogSobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied.
Functional inequalities for heavy tails distributions and application to isoperimetry
, 2008
"... Abstract. This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and th ..."
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Cited by 5 (4 self)
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Abstract. This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures onR n we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve many previous results.
Weak logarithmic Sobolev inequalities and entropic convergence
, 2005
"... In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion ..."
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Cited by 4 (2 self)
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In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semigroup. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.
On the convexity of the entropy along entropic interpolations
"... Abstract. Convexity properties of the entropy along displacement interpolations are crucial in the LottSturmVillani theory of lower bounded curvature of geodesic measure spaces. As discrete spaces fail to be geodesic, an alternate analogous theory is necessary in the discrete setting. Replacing di ..."
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Cited by 1 (0 self)
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Abstract. Convexity properties of the entropy along displacement interpolations are crucial in the LottSturmVillani theory of lower bounded curvature of geodesic measure spaces. As discrete spaces fail to be geodesic, an alternate analogous theory is necessary in the discrete setting. Replacing displacement interpolations by entropic ones allows for developing a rigorous calculus, in contrast with Otto’s informal calculus. When the underlying state space is a Riemannian manifold, we show that the first and second derivatives of the entropy as a function of time along entropic interpolations are expressed in terms of the standard BakryÉmery operators Γ and Γ2. On the other hand, in the discrete setting new operators appear. Our approach is probabilistic; it relies on the Markov property and time reversal. We illustrate these calculations by means of Brownian diffusions on manifolds and random walks on graphs. We also give a new unified proof, covering both the manifold and graph cases, of a logarithmic Sobolev inequality in connection with convergence to equilibrium.
Higher Order Entropies
"... Higher order entropies are kinetic entropy estimators for fluid models. These quantities are quadratics in the velocity and temperature derivatives and have temperature dependent coefficients. We investigate governing equations for higher order entropies and related a priori estimates in the natural ..."
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Higher order entropies are kinetic entropy estimators for fluid models. These quantities are quadratics in the velocity and temperature derivatives and have temperature dependent coefficients. We investigate governing equations for higher order entropies and related a priori estimates in the natural situation where viscosity and thermal conductivity depend on temperature. We establish conditionnal entropicity, that is, positivity of higher order derivatives source terms in these governing equations provided that ‖ log T ‖BMO + ‖v / √ T ‖L ∞ is small enough. The temperature factors renormalizing temperature and velocity derivatives then yield majorization of lower order convective terms only when the temperature dependence of transport coefficients is taken into account according to the kinetic theory. In this situation, we obtain entropic principles for higher order entropies of arbitrary order. As an application, we investigate a priori estimates and global existence of solutions when the initial values log(T0/T∞) and v0 / √ T0 are small enough in appropriate spaces.
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"... trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations J.Fontbona ∗ B.Jourdain † ..."
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trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations J.Fontbona ∗ B.Jourdain †
Poincaré inequality and the Lp . . .
, 2010
"... We prove that for symmetric Markov processes of diffusion type admitting a “carré du champ”, the Poincaré inequality is equivalent to the exponential convergence of the associated semigroup in one (resp. all) L p (µ) spaces for 1 < p < +∞. Part of this result extends to the stationary non n ..."
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We prove that for symmetric Markov processes of diffusion type admitting a “carré du champ”, the Poincaré inequality is equivalent to the exponential convergence of the associated semigroup in one (resp. all) L p (µ) spaces for 1 < p < +∞. Part of this result extends to the stationary non necessarily symmetric situation.