Regularization in L1 for the Ornstein-Uhlenbeck semigroup
Lehec, Joseph1
1 CEREMADE (UMR CNRS 7534) Université Paris–Dauphine.
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 1, pp. 191-204.

Soit γn la mesure Gaussienne standard sur ℝn et soit (Qt) le semi-groupe d’Ornstein-Uhlenbeck. Eldan et Lee ont montré récemment que pour toute fonction positive f d’intégrale 1 et pour temps t la queue de distribution de Qtf vérifie

γn({Qtf>r})≤Ct(loglogr)4rlogr,∀r>1

où Ct est une constante dépendant seulement de t et pas de la dimension. L’objet de cet article est de simplifier en partie leur démonstration et d’éliminer le facteur (loglogr)4.

Let γn be the standard Gaussian measure on ℝn and let (Qt) be the Ornstein-Uhlenbeck semigroup. Eldan and Lee recently established that for every non-negative function f of integral 1 and any time t the following tail inequality holds true:

γn({Qtf>r})≤Ct(loglogr)4rlogr,∀r>1

where Ct is a constant depending on t but not on the dimension. The purpose of the present paper is to simplify parts of their argument and to remove the (loglogr)4 factor.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1492
Lehec, Joseph 1

1 CEREMADE (UMR CNRS 7534) Université Paris–Dauphine. 
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Lehec, Joseph. Regularization in $L_1$ for the Ornstein-Uhlenbeck semigroup. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 1, pp. 191-204. doi : 10.5802/afst.1492. https://www.numdam.org/articles/10.5802/afst.1492/

[1] Ball (K.), Barthe (F.), Bednorz (W.), Oleszkiewicz (K.), Wolff (P.).— L1-smoothing for the Ornstein-Uhlenbeck semigroup, Mathematika 59, no. 1, p. 160-168 (2013). | DOI | Zbl

[2] Eldan (R.), Lee (J.).— Regularization under diffusion and anti-concentration of temperature, arXiv:1410.3887. | DOI | MR

[3] Gross (L.).— Logarithmic Sobolev inequalities, Amer. J. Math. 97, no. 4, p. 1061-1083 (1975). | DOI | MR | Zbl

[4] Lehec (J.).— Representation formula for the entropy and functional inequalities, Ann. Inst. Henri Poincaré Probab. Stat. 49, no. 3, p. 885-899 (2013). | DOI | Numdam | MR | Zbl

[5] Liptser (R.), Shiryaev (A.).— Statistics of random processes., Vol I, general theory. 2nd edition, Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin (2001). | DOI

[6] Nelson (E.).— The free Markoff field. J. Functional Analysis 12, p. 211-227 (1973). | DOI | MR | Zbl

[7] Talagrand (M.).— A conjecture on convolution operators, and a non-Dunford-Pettis operator on L1, Israel J. Math. 68, no. 1, p. 82-88 (1989). | DOI | Zbl

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