Transportation inequalities for stochastic differential equations of pure jumps
type: Item
instance of: scholarly article
title: Transportation inequalities for stochastic differential equations of pure jumps
zbMATH Open document ID: 1209.60015
DOI: 10.1214/09-AIHP320
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2010-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/241053
review text: The law of stochastic diffusion equations with constant variance coefficient is known to satisfy a Poincaré variance inequality that shows the exponential decay of the associated semigroup under the invariant probability measure of the equation solution. This Poincaré inequality is a consequence of a transportation inequality which itself follows from the a logarithmic Sobolev inequality. An extension of these results has been given by the author and co-authors to the setting of a non-constant variance coefficient under the form of a transportation inequality that holds under a dissipativity condition. The goal of this paper is to extend such results to stochastic differential equations of pure jumps, for which the Poincaré inequality does not hold in general, by deriving \(W_1 H\) transportation inequalities under the \(L^1\) and \(L^\infty\)-Wasserstein distances. Concentration and deviation inequalities of Poisson type for the equation solution are obtained as a consequence. The proofs rely on a bound on the Laplace transform of the solution obtained by forward-backward stochastic calculus and the Clark representation formula of the Malliavin calculus for jump processes on the one hand, and on an equivalence result by \textit{N. Gozlan} and \textit{C. Léonard} [Probab. Theory Relat. Fields 139, No. 1--2, 235--283 (2007; Zbl 1126.60022)] between Laplace bounds and \(W_1 H\) transportation inequalities for the Wasserstein distance on the other hand.
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 5758857
zbMATH Keywords: transportation inequalities
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: jump processes
zbMATH Keywords: Malliavin calculus
author: Liming Wu
MaRDI profile type: Publication
cites work: Q5186515
cites work: Q3996311
cites work: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
cites work: Quantitative concentration inequalities for empirical measures on non-compact spaces
cites work: Transportation cost-information inequalities and applications to random dynamical systems and diffusions.
cites work: Q5503475
cites work: A large deviation approach to some transportation cost inequalities
cites work: Transportation-information inequalities for Markov processes
cites work: The Euler scheme for Lévy driven stochastic differential equations: limit theorems.
cites work: Convex concentration inequalities and forward-backward stochastic calculus
cites work: The concentration of measure phenomenon
cites work: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality
cites work: Duality formulas on the Poisson space
cites work: Q4805362
cites work: Harnack and functional inequalities for generalized Mehler semigroups.
cites work: A new modified logarithmic Sobolev inequality for Poisson point processes and several applications
cites work: Uniformly integrable operators and large deviations for Markov processes
cites work: Essential spectral radius for Markov semigroups. I: Discrete time case
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description: scientific article; zbMATH DE number 5758857