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Psi-entropy inequalities for the Ornstein-Uhlenbeck semigroup

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Abstract

We study the Ornstein-Uhlenbeck semigroup (P t ) t≥0={exp(tL)} t≥0 generated by the operator Lf(x)=Δf(x)−x⋅∇f(x), on ℝn equipped with the n-dimensional standard Gaussian measure \(\gamma_{n}(\mathrm{d}x)=(\sqrt{2\pi})^{-n}\exp(-|x|^{2}/2)\mathrm{d}x\). By means of a simple method involving essentially a commutation property between the semigroup and the gradient, we describe a large family of optimal integral inequalities with the logarithmic Sobolev and reverse logarithmic Sobolev inequalities as particular cases.

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Acknowledgements

We wish to thank Professor Fouad Zitan for his encouragements and helpful comments. The authors are indebted to the referee for the valuable comments, remarks and suggestions, which improve the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, Faculty of Sciences, Équipe d’Analyse Harmonique et Probabilités University Moulay Ismaïl, BP 11201, Zitoune, Meknès, Morocco

    Abdellatif Bentaleb, Saïd Fahlaoui & Ali Hafidi

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  1. Abdellatif Bentaleb
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  2. Saïd Fahlaoui
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  3. Ali Hafidi
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Corresponding author

Correspondence to Abdellatif Bentaleb.

Additional information

Communicated by Jerome A. Goldstein.

This paper was written while A. Bentaleb was visiting the International Centre for Theoretical Physics, Trieste (Italy) in October 2011.

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Bentaleb, A., Fahlaoui, S. & Hafidi, A. Psi-entropy inequalities for the Ornstein-Uhlenbeck semigroup. Semigroup Forum 85, 361–368 (2012). https://doi.org/10.1007/s00233-012-9421-3

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  • DOI: https://doi.org/10.1007/s00233-012-9421-3

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