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Restricted hypercontractivity on the Poisson space


Authors: Ivan Nourdin, Giovanni Peccati and Xiaochuan Yang
Journal: Proc. Amer. Math. Soc. 148 (2020), 3617-3632
MSC (2010): Primary 60H07, 60E15, 60E05
DOI: https://doi.org/10.1090/proc/14964
Published electronically: May 8, 2020
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Abstract: We show that the Ornstein-Uhlenbeck semigroup associated with a general Poisson random measure is hypercontractive, whenever it is restricted to non-increasing mappings on configuration spaces. We deduce from this result some versions of Talagrand's $ L^1$-$ L^2$ inequality for increasing and concave mappings, and we build examples showing that such an estimate represents a strict improvement of the classical Poincaré inequality. We complement our finding with several results of independent interest, such as gradient estimates and an inequality with isoperimetric content.


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Additional Information

Ivan Nourdin
Affiliation: Department of Mathematics, Université de Luxembourg, Maison du Nombre, 6 avenue de la Fonte, L-4364 Esch-sur-Alzette, Grand Duchy of Luxembourg
Email: ivan.nourdin@uni.lu

Giovanni Peccati
Affiliation: Department of Mathematics, Université de Luxembourg, Maison du Nombre, 6 avenue de la Fonte, L-4364 Esch-sur-Alzette, Grand Duchy of Luxembourg
Email: giovanni.peccati@uni.lu

Xiaochuan Yang
Affiliation: Department of Mathematics, Université de Luxembourg, Maison du Nombre, 6 avenue de la Fonte, L-4364 Esch-sur-Alzette, Grand Duchy of Luxembourg
Email: xiaochuan.yang@uni.lu

DOI: https://doi.org/10.1090/proc/14964
Keywords: Hypercontractivity, improved Poincar\'e inequality, modified logarithmic Sobolev inequality, Poisson processes, superconcentration.
Received by editor(s): April 17, 2019
Received by editor(s) in revised form: November 27, 2019
Published electronically: May 8, 2020
Additional Notes: The first author was supported by the FNR grant APOGee at Luxembourg University.
The second author was supported by the FNR grant FoRGES (R-AGR-3376-10) at Luxembourg University.
The third author was supported by the FNR Grant MISSILe (R-AGR-3410-12-Z) at Luxembourg and Singapore Universities.
Communicated by: Zhen-Qing Chen
Article copyright: © Copyright 2020 American Mathematical Society