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Stochastic calculus and degenerate boundary value problems

Patrick Cattiaux — 1992

Annales de l'institut Fourier

Consider the boundary value problem (L.P): ( h - A ) u = f in D , ( v - Γ ) u = g on D where A is written as A = 1 / 2 i = 1 m Y i 2 + Y 0 , and Γ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields Y i ( 0 i m ), for regular open sets D with a non-characteristic boundary. Our study lies on the stochastic representation of u and uses the stochastic calculus of variations for the ( A , Γ ) -diffusion...

Deviation bounds for additive functionals of Markov processes

Patrick CattiauxArnaud Guillin — 2008

ESAIM: Probability and Statistics

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.).

Trends to equilibrium in total variation distance

Patrick CattiauxArnaud Guillin — 2009

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

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 ψ ....

deviation bounds for additive functionals of markov processes

Patrick CattiauxArnaud Guillin — 2007

ESAIM: Probability and Statistics

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 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 Poincaré etc.).

Limit theorems for some functionals with heavy tails of a discrete time Markov chain

Patrick CattiauxMawaki Manou-Abi — 2014

ESAIM: Probability and Statistics

Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain ( ≥ 0) with invariant distribution . We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional S n = i = 1 n f ( X i ) S n = ∑ i = 1 n f ( X i ) for a possibly non square integrable function. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing...

Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry.

Franck BarthePatrick CattiauxCyril Roberto — 2006

Revista Matemática Iberoamericana

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general F-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the...

Poincaré inequalities and hitting times

Patrick CattiauxArnaud GuillinPierre André Zitt — 2013

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

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions is well known. We give here the correspondence (with quantitative results) for reversible diffusion processes. As a consequence, we generalize results of Bobkov in the one dimensional case on the value of the Poincaré constant for log-concave measures to superlinear potentials. Finally, we study various functional inequalities under different hitting times integrability conditions (polynomial,…)....

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