Displaying similar documents to “Large deviation principles for Markov processes via phi-Sobolev inequalities.”

Deviation bounds for additive functionals of Markov processes

Patrick Cattiaux, Arnaud Guillin (2008)

ESAIM: Probability and Statistics

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In this paper we derive non asymptotic deviation bounds for ν ( | 1 t ∫ 0 t V ( X s ) d s - ∫ V d μ | ≥ R ) where X is a μ stationary and ergodic Markov process and V is some μ integrable function. These bounds are obtained under various moments assumptions for V , and various regularity assumptions for μ . Regularity means here that μ may satisfy various functional inequalities (F-Sobolev, generalized Poincaré etc.).

From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality

Ivan Gentil (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

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We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux in [BL00]. Using the Prékopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on ℝ n , with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality, developed in [GGM05, GGM07], for all uniformly strictly convex potential as well as the Euclidean logarithmic Sobolev inequality.

On equivalence of super log Sobolev and Nash type inequalities

Marco Biroli, Patrick Maheux (2014)

Colloquium Mathematicae

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We prove the equivalence of Nash type and super log Sobolev inequalities for Dirichlet forms. We also show that both inequalities are equivalent to Orlicz-Sobolev type inequalities. No ultracontractivity of the semigroup is assumed. It is known that there is no equivalence between super log Sobolev or Nash type inequalities and ultracontractivity. We discuss Davies-Simon's counterexample as the borderline case of this equivalence and related open problems.

Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions

Radosław Adamczak, Michał Strzelecki (2015)

Studia Mathematica

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We provide a mild sufficient condition for a probability measure on the real line to satisfy a modified log-Sobolev inequality for convex functions, interpolating between the classical log-Sobolev inequality and a Bobkov-Ledoux type inequality. As a consequence we obtain dimension-free two-level concentration results for convex functions of independent random variables with sufficiently regular tail decay. We also provide a link between modified log-Sobolev...

Trends to equilibrium in total variation distance

Patrick Cattiaux, Arnaud Guillin (2009)

Annales de l'I.H.P. Probabilités et statistiques

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This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound “à la Pinsker” enabling us to study our problem firstly via usual functional inequalities (Poincaré inequality, weak Poincaré,…) and truncation procedure, and secondly through the introduction of new functional inequalities...