Consider the boundary value problem (L.P): in , on where is written as , and 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 (), for regular open sets with a non-characteristic boundary.
Our study lies on the stochastic representation of and uses the stochastic calculus of variations for the -diffusion...
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 Poincaré etc.).
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 ....
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 Poincaré etc.).
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 ) 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...
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...
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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