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The theory of one-mode type Interacting Fock Space (IFS) allows us to construct the quantum decomposition associated with stochastic processes on R with moments of any order. The problem to extend this result to processes without moments of any order is still open but the Araki-Woods-Parthasarathy-Schmidt characterization of Lévy processes in terms of boson Fock spaces, canonically associated with the Lévy-Khintchine functions of these processes, provides a quantum decomposition for tchem which is based on boson creations, annihilation and preservation operators rather than on their IFS counterparts. In order to compare the two quantum decompositions in their common domain of application (i.e., the Lévy processes with moments of all orders) the first step is to give a precise formulation of the quantum decomposition for these processes and the analytical conditions of its validity. We show that these conditions distinguish three different notions of quantum decomposition of a Lévy process on R according to the existence of second or only first moments, or no moments at all. For the last class a multiplicative renormalization procedure is needed.
Czasopismo
Rocznik
Tom
Strony
337--362
Opis fizyczny
Bibliogr. 26 poz.
Twórcy
autor
- Centro Vito Volterra, Facultà di Economia, Università di Tor Vergata, Via di Tor Vergata, 00133 Roma, Italy
autor
- Department of Mathematics, Graduate School of Science and Technology of Hammam Sousse University of Sousse, Tunisia
autor
- Department of Mathematics, Higher Institute of Applied Science and Technology of Gabes University of Gabes, Tunisia
Bibliografia
- [1] L. Accardi, A. Barhoumi, and A. Riahi, White noise Lévy-Meixner processes through a transfer principle from one-mode to one-mode type interacting Fock spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), pp. 435-460.
- [2] L. Accardi and M. Bożejko, Interacting Fock space and Gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), pp. 663-670.
- [3] L. Accardi, H.-H. Kuo, and A. I. Stan, Characterization of probability measures through the canonically associated interacting Fock spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (4) (2004), pp. 485-505.
- [4] L. Accardi, H. Ouerdiane, and H. Rebei, Lévy processes through time shift on oscillator Weyl algebra, Commun. Stoch. Anal. 6 (1) (2012), pp. 125-155.
- [5] L. Accardi, H. Rebei, and A. Riahi, The quantum decomposition of random variables without moments, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2) (2013), 1350012.
- [6] S. Albeverio, Yu. G. Kondratiev, and L. Streit, How to generalize white noise analysis to non-Gaussian spaces, in: Dynamics of Complex and Irregular Systems, Ph. Blanchard, L. Streit, M. Sirugue-Collin, and D. Testard (Eds.),World Scientific, Singapore 1993, pp. 120-130.
- [7] H. Araki, Factorizable representation of current algebra: Non-commutative extension of the Lévy-Kinchin formula and cohomology of a solvable group with values in a Hilbert space, Publ. Res. Inst. Math. Sci., Kyoto Univ. 5 (1969/70), pp. 361-422.
- [8] Yu. M. Berezansky and D. A. Mierzejewski, The construction of the chaotic representation for the gamma field, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), pp. 33-56.
- [9] G. Di Nunno, B. Øksendal, and F. Proskea, White noise analysis for Lévy processes, J. Funct. Anal. 206 (2004), pp. 109-148.
- [10] A. Guichardet, Symmetric Hilbert Spaces and Related Topics, Lecture Notes in Math., Vol. 261, Springer, Berlin 1972.
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- [12] Z. Y. Huang and Y. Wu, Lévy white noise calculus based on interaction exponents, Acta Appl. Math. 88 (2005), pp. 251-268.
- [13] Y. Itô, Generalized Poisson functionals, Probab. Theory Related Fields 77 (1988), pp. 1-28.
- [14] Y. Itô and I. Kubo, Calculus on Gaussian and Poisson white noises, Nagoya Math. J. 111 (1988), pp. 41-84.
- [15] A. N. Kolmogorov, Kurven in Hilbertschen Raum, die gegenüber einer einparametrigen Gruppe von Bewegungen invariant sind, Dokl. Akad. Nauk SSSR 26 (1940), pp. 6-9.
- [16] A. N. Kolmogorov, Stationary sequences in Hilbert space, Moscow Univ. Math. Bull. 2 (1941), pp. 1-40.
- [17] Yu. G. Kondratiev and E. W. Lytvynov, Operators of gamma white noise calculus, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), pp. 303-335.
- [18] Yu. G. Kondratiev, J. L. da Silva, L. Streit, and G. F. Us, Analysis on Poisson and gamma spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), pp. 91-117.
- [19] Y.-J. Lee and H.-H. Shih, Analysis of generalized Lévy white noise functionals, J. Funct. Anal. 211 (2004), pp. 1-70.
- [20] Y.-J. Lee and H.-H. Shih, Lévy white noise measures on infinite-dimensional spaces: Existence and characterization of the measurable support, J. Funct. Anal. 237 (2006), pp. 617-633.
- [21] K. R. Parthasarathy, A quantum stochastic approach to Itô’s formula for Lévy processes, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), pp. 1417-1420.
- [22] K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel 1992.
- [23] K. R. Parthasarathy and K. Schmidt, Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory, Lecture Notes in Math., Vol. 272, Springer, 1972.
- [24] M. Reed and B. Simon, Functional Analysis, Methods of Modern Mathematical Physics, Vol. 1, Academic Press, 1980.
- [25] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Stud. Adv. Math., Vol. 68, Cambridge University Press, Cambridge 1999.
- [26] A. M. Vershik and N. V. Tsilevich, Fock factorizations, and decompositions of the L2 spaces over general Lévy processes, Uspekhi Mat. Nauk 58 (3) (2003), pp. 3-50. English translation: Russian Math. Surveys 58 (3) (2003), pp. 427-472.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
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