104 research outputs found
Hypercontractivity for perturbed diffusion semigroups
being a nonnegative measure satisfying some log-Sobolev inequality, we
give conditions on F for the measure to also satisfy some
log-Sobolev inequality. Explicit examples are studied
A criterion for Talagrand's quadratic transportation cost inequality
We show that the quadratic transportation cost inequality is equivalent
to both a Poincar\'e inequality and a strong form of the Gaussian concentration
property. The main ingredient in the proof is a new family of inequalities,
called modified quadratic transportation cost inequalities in the spirit of the
modified logarithmic-Sobolev inequalities by Bobkov and Ledoux \cite{BL97},
that are shown to hold as soon as a Poincar\'e inequality is satisfied
Functional Inequalities via Lyapunov conditions
We review here some recent results by the authors, and various coauthors, on
(weak,super) Poincar\'e inequalities, transportation-information inequalities
or logarithmic Sobolev inequality via a quite simple and efficient technique:
Lyapunov conditions
Deviation bounds for additive functionals of Markov process
In this paper we derive non asymptotic deviation bounds for where is a stationary and ergodic Markov process and is
some integrable function. These bounds are obtained under various moments
assumptions for , and various regularity assumptions for . Regularity
means here that may satisfy various functional inequalities (F-Sobolev,
generalized Poincar\'e etc...)
Trends to Equilibrium in Total Variation Distance
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 "\`{a} la Pinsker"
enabling us to study our problem firstly via usual functional inequalities
(Poincar\'{e} inequality, weak Poincar\'{e},...) and truncation procedure, and
secondly through the introduction of new functional inequalities \Ipsi. These
\Ipsi-inequalities are characterized through measure-capacity conditions and
-Sobolev inequalities. A direct study of the decay of Hellinger distance is
also proposed. Finally we show how a dynamic approach based on reversing the
role of the semi-group and the invariant measure can lead to interesting
bounds.Comment: 36 page
Rate of Converrgence for ergodic continuous Markov processes : Lyapunov versus Poincare
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 Poincar\'e type). We show that they can be linked through new
inequalities (Lyapunov-Poincar\'e inequalities). Explicit examples for
diffusion processes are studied, improving some results in the literature. The
example of the kinetic Fokker-Planck equation recently studied by H\'erau-Nier,
Helffer-Nier and Villani is in particular discussed in the final section
Concentration for independent random variables with heavy tails
If a random variable is not exponentially integrable, it is known that no
concentration inequality holds for an infinite sequence of independent copies.
Under mild conditions, we establish concentration inequalities for finite
sequences of independent copies, with good dependence in
Isoperimetry between exponential and Gaussian
We study in details the isoperimetric profile of product probability measures
with tails between the exponential and the Gaussian regime. In particular we
exhibit many examples where coordinate half-spaces are approximate solutions of
the isoperimetric problem
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