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WICK CALCULUS FOR THE SQUARE OF A GAUSSIAN RANDOM VARIABLE WITH APPLICATION TO YOUNG AND HYPERCONTRACTIVE INEQUALITIES

ALBERTO LANCONELLI

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italia

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and

LUIGI SPORTELLI

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italia

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https://doi.org/10.1142/S021902571250018XCited by:3 (Source: Crossref)

Abstract

We investigate a probabilistic interpretation of the Wick product associated to the chi-square distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian⁷ and Poissonian¹² cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem.

Keywords:

AMSC: 33C45, 46F10, 60E05

References

L. Accardi, Y. G. Lu and I. Volovich, Acta Appl. Math. 63, 3 (2000), DOI: 10.1023/A:1010719902298. Crossref, Web of Science, Google Scholar
N. I. Akhiezer , The Classical Moment Problem and Some Related Questions in Analysis ( Oliver & Boyd Ltd , 1965 ) . Google Scholar
N. Asai, I. Kubo and H.-H. Kuo, Proceed. Amer. Math. Soc. 131, 815 (2002), DOI: 10.1090/S0002-9939-02-06564-4. Crossref, Web of Science, Google Scholar
D. Bakry and M. Emery, C. R. Acad. Sci. Paris, t 299(15), 775 (1984). Web of Science, Google Scholar
F. Biagini et al. , Stochastic Calculus for Fractional Brownian Motion and Applications ( Springer , 2008 ) . Crossref, Google Scholar
T. S. Chihara , An Introduction to Orthogonal Polynomials ( Gordon and Breach, Science Publishers Inc , 1978 ) . Google Scholar
P. Da Pelo, A. Lanconelli and A. I. Stan, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14, 375 (2011), DOI: 10.1142/S0219025711004456. Link, Web of Science, Google Scholar
P. Deift , Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , Courant Lecture Notes in Mathematics ( American Mathematical Society , 2000 ) . Crossref, Google Scholar
H. Holden et al. , Stochastic Partial Differential Equations — A Modeling, White Noise Functional Approach ( Birkhäuser , Boston , 1996 ) . Crossref, Google Scholar
S. Janson , Gaussian Hilbert spaces , Cambridge Tracts in Mathematics 129 ( Cambridge University Press , 1997 ) . Crossref, Google Scholar
H.-H. Kuo , White Noise Distribution Theory , Probability and Stochastic Series ( CRC Press, Inc , 1996 ) . Google Scholar
A. Lanconelli and L. Sportelli, On a connection between the Poissonian Wick product and the discrete convolution, To appear on Communications on Stochastic Analysis (2011) . Google Scholar
A. Lanconelli and A. I. Stan, J. Appl. Math. Stoch. Anal. 22 (2008), DOI: 10.1155/2008/254897. Google Scholar
J. Potthoff and M. Timpel, Potential Analysis 4, 637 (1995), DOI: 10.1007/BF02345829. Crossref, Web of Science, Google Scholar
A. I. Stan, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11, 377 (2008), DOI: 10.1142/S0219025708003178. Link, Web of Science, Google Scholar
G. Szegö, Orthogonal Polynomials, 4th edn. (American Mathematical Society, Providence, Rhode Island, 1975). Google Scholar
G. C. Wick, Phys. Rev. 80, 268 (1950), DOI: 10.1103/PhysRev.80.268. Crossref, Web of Science, Google Scholar