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A note on Gamma Wick products

Florin Catrina

Department of Mathematics and Computer Science, St. John’s University, Queens, NY 11439, USA

E-mail Address: catrinaf@stjohns.edu

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and

Aurel I. Stan

Department of Mathematics, Ohio State University at Marion, 1465 Mount Vernon Avenue, Marion, OH 43302, USA

E-mail Address: stan.7@osu.edu

Corresponding author.

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https://doi.org/10.1142/S0219025718500042Cited by:1 (Source: Crossref)

Abstract

An integral representation of the Wick product for Gamma distributed random variables, with mean greater than $1 / 2$ , is presented first. We use this integral representation to prove a Hölder inequality for norms of Gamma Wick products.

Communicated by L. Accardi

Keywords:

AMSC: 33C45, 46F12, 60H10, 60H40

References

1. W. Ayed and H.-H. Kuo, An extension of the Itô integral, Commun. Stochast. Anal. 2 (2008) 323–333. Google Scholar
2. W. Ayed and H.-H. Kuo, An extension of the Itô integral: Toward a general theory of stochastic integration, Theory Stochast. Process. 16 (2010) 17–28. Google Scholar
3. D. Bakry and M. Emery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris I 299 (1984) 775–778. Google Scholar
4. W. Beckner, Inequalities in Fourier analysis, Ann. of Math. 102 (1975) 159–182. Crossref, Web of Science, Google Scholar
5. J. Bennett, A. Carbery, M. Christ and T. Tao, The Brascamp–Lieb inequalities: Finiteness, structure and extremals, preprint (2005), arXiv:math/0505065. Google Scholar
6. H. J. Brascamp and E. H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Adv. Math. 20 (1976) 151–173. Crossref, Web of Science, Google Scholar
7. E. Carlen and D. Cordero-Erausquin, Subadditivity of the entropy and its relation to Brascamp–Lieb type inequalities, Geom. Funct. Anal. 19 (2009) 373–405. Crossref, Web of Science, Google Scholar
8. E. Carlen, E. Lieb and M. Loss, A sharp analog of Young’s inequality on $S^{N}$ and related entropy inequalities, J. Geom. Anal. 14 (2004) 487–520. Crossref, Web of Science, Google Scholar
9. P. Da Pelo, A. Lanconelli and A. I. Stan, A Hölder–Young–Lieb inequality for norms of Gaussian Wick products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 (2011) 375–407. Link, Web of Science, Google Scholar
10. P. Da Pelo and A. Lanconelli, On a new probabilistic representation for the solution of the heat equation, Stochastics 84 (2012) 171–181. Crossref, Google Scholar
11. P. Da Pelo, A. Lanconelli and A. I. Stan, An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems, Stochast. Process Appl. 123 (2013) 3183–3200. Crossref, Web of Science, Google Scholar
12. P. Da Pelo, A. Lanconelli and A. I. Stan, A sharp interpolation between the Hölder and Gaussian Young inequalities, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19 (2016), Article ID: 1650001, pp. 37. https://dx.doi.org/10.1142/S0219025716500016. Link, Web of Science, Google Scholar
13. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Rev. edn., Textbooks in Mathematics (CRC Press, 2015). Crossref, Google Scholar
14. L. Gross, Logarithmic Sobolev inequality, Amer. J. Math. 97 (1975) 1061–1083. Crossref, Web of Science, Google Scholar
15. H.-H. Kuo, K. Saitô and A. I. Stan, A Hausdorff–Young inequality for white noise analysis, in Quantum Information IV, eds. T. Hida and K. Saitô (World Scientific, 2002), pp. 115–126. Link, Google Scholar
16. A. Lanconelli and L. Sportelli, Wick calculus for the square of a Gaussian random variable with application to Young and hypercontractive inequalities, Intin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (2012), Article ID: 1250018, 16 pp. Web of Science, Google Scholar
17. A. Lanconelli and A. I. Stan, Hölder type inequalities for norms of Wick products, J. Appl. Math. Stochast. Anal. 2008 (2008), Article ID: 254897, 22 pp. https://doi.org/10.1155/2008/254897. Crossref, Google Scholar
18. A. Lanconelli and A. I. Stan, Some inequalities for norms of Gaussian Wick products, Stochast. Anal. Appl. 28 (2010) 523–539. Crossref, Web of Science, Google Scholar
19. A. Lanconelli and A. I. Stan, A Hölder inequality for norms of Poissonian Wick products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013), Article ID: 1350022, 39 pp. Link, Web of Science, Google Scholar
20. E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990) 179–208. Crossref, Web of Science, Google Scholar
21. E. H. Lieb and M. Loss, Analysis, 2nd edn., Graduate Studies in Mathematics, Vol. 14 (Amer. Math. Soc., 2001). Crossref, Google Scholar
22. E. Nelson, The free Markoff field, J. Funct. Anal. 12 (1973) 211–227. Crossref, Google Scholar