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Publisher:
Cambridge University Press
Online publication date:
February 2016
Print publication year:
2016
Online ISBN:
9781316415054

Book description

This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus.

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Contents

References
[1] L, Accardi, U., Franz, and M., Skeide.Renormalized squares of white noise and other non-Gaussian noises as Lévy processes on real Lie algebras. Comm. Math. Phys., 228(1):123–150, 2002. (Cited on pages xvii, 17, 38, 40, 96, 132, 147, and 187).
[2] L., Accardi, M., Schürmann, and W.v., Waldenfels.Quantum independent increment processes on superalgebras. Math. Z., 198:451–477, 1988. (Cited on pages xvii and 131).
[3] G.S., Agarwal.|Quantum Optics. Cambridge University Press, Cambridge, 2013. (Cited on page 130).
[4] N.I., Akhiezer.The Classical Moment Problem and Some Related Questions in Analysis. Translated by N., Kemmer.Hafner Publishing Co., New York, 1965. (Cited on page 54).
[5] N.I., Akhiezer and I.M., Glazman.Theory of Linear Operators in Hilbert Space. Dover Publications Inc., New York, 1993. (Cited on page 243).
[6] S., Albeverio, Yu. G., Kondratiev, and M., Röckner.Analysis and geometry on configuration spaces.J. Funct. Anal., 154(2):444–500, 1998. (Cited on page 162).
[7] S.T., Ali, N.M., Atakishiyev, S.M., Chumakov, and K.B., Wolf.The Wigner function for general Lie groups and the wavelet transform.Ann. Henri Poincaré, 1(4):685–714, 2000. (Cited on pages xvi, xvii, 114, 115, 118, 120, 122, 123, 124, and 191).
[8] S.T., Ali, H., Führ, and A.E., Krasowska.Plancherel inversion as unified approach to wavelet transforms and Wigner functions. Ann. Henri Poincaré, 4(6):1015– 1050, 2003. (Cited on pages xvi and 124).
[9] G.W., Anderson, A., Guionnet, and O., Zeitouni.An Introduction to Random Matrices. Cambridge: Cambridge University Press, 2010. (Cited on page 88).
[10] U., Franz and A., Skalski. Noncommutative Mathematics for Quantum Systems. To appear in the Cambridge IISc Series, 2015. D. Applebaum, Probability on compact Lie groups, volume 70 of Probability Theory and Stochastic Modelling, Springer, 2014” after 45th entry (Cited on pages xvi and 99).
[11] M., Anshelevich. Orthogonal polynomials and counting permutations. www .math.tamu.edu/~manshel/papers/OP-counting-permutations.pdf, 2014. (Cited on page 233).
[12] N.M., Atakishiyev, S.M., Chumakov, and K.B., Wolf.Wigner distribution function for finite systems. J. Math. Phys., 39(12):6247–6261, 1998. (Cited on page 125).
[13] V.P., Belavkin.A quantum nonadapted Itô formula and stochastic analysis in Fock scale. J. Funct. Anal., 102:414–447, 1991. (Cited on pages 88, 190, 212, and 213).
[14] V.P., Belavkin.A quantum nonadapted stochastic calculus and nonstationary evolution in Fock scale. In Quantum Probability and Related Topics VI, pages 137–179. World Sci. Publishing, River Edge, NJ, 1991. (Cited on pages 88, 190, 212, and 213).
[15] V.P., Belavkin.On quantum Itô algebras. Math. Phys. Lett., 7:1–16, 1998. (Cited on page 88).
[16] C., Berg, J.P.R., Christensen, and P., Ressel.Harmonic Analysis on Semigroups, volume 100 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1984. Theory of positive definite and related functions. (Cited on page 88).
[17] Ph., Biane.Calcul stochastique non-commutatif. In Ecole d'Eté de Probabilités de Saint-Flour, volume 1608 of Lecture Notes in Mathematics.Springer-Verlag, Berlin, 1993. (Cited on pages 7, 88, 211, and 264).
[18] Ph., Biane.Quantum Markov processes and group representations. In Quantum Probability Communications, QP-PQ, X, pages 53–72. World Sci. Publishing, River Edge, NJ, 1998. (Cited on pages xvii, 132, and 189).
[19] Ph., Biane and R., Speicher.Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Related Fields, 112(3):373–409, 1998. (Cited on pages 88 and 216).
[20] L.C., Biedenharn and J.D., Louck.Angular Momentum in Quantum Physics. Theory and Application. With a foreword by P.A. Carruthers. Cambridge: Cambridge University Press, reprint of the 1981 hardback edition edition, 2009. (Cited on page 72).
[21] L.C., Biedenharn and J.D., Louck.The Racah-Wigner Algebra in Quantum Theory. With a foreword by P.A. Carruthers. Introduction by G.W. Mackey.Cambridge: Cambridge University Press, reprint of the 1984 hardback ed. edition, 2009. (Cited on page 72).
[22] J.-M., Bismut.Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions. Z. Wahrsch. Verw. Gebiete, 56(4):469–505, 1981. (Cited on pages xvii and 189).
[23] F., Bornemann.Teacher's corner - kurze Beweise mit langer Wirkung. Mitteilungen der Deutschen Mathematiker-Vereinigung, 10:55–55, July 2002. (Cited on page 258).
[24] N., Bouleau, editor. Dialogues autour de la création mathématique.Association Laplace-Gauss, Paris, 1997.
[25] T., Carleman.Les Fonctions Quasi Analytiques. Paris: Gauthier-Villars, Éditeur, Paris, 1926. (Cited on page 221).
[26] M.H., Chang.Quantum Stochastics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2015. (Cited on pages xviii and 88).
[27] T.S., Chihara.An Introduction to Orthogonal Polynomials. Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and Its Applications, Vol. 13. (Cited on page 233).
[28] S.M., Chumakov, A.B., Klimov, and K.B., Wolf.Connection between two wigner functions for spin systems. Physical Review A, 61(3):034101, 2000. (Cited on page 125).
[29] L., Cohen.Time-Frequency Analysis: Theory and Applications. Prentice-Hall, New Jersey, 1995. (Cited on pages xvii and 130).
[30] M., Cook.Mathematicians: An Outer View of the Inner World. Princeton University Press, USA, 2009. With an introduction by R. C. Gunning.
[31] D., Dacunha-Castelle and M., Duflo.Probability and Statistics. Vol. I. Springer- Verlag, New York, 1986. (Cited on page xvii).
[32] D., Dacunha-Castelle and M., Duflo.Probability and Statistics. Vol. II. Springer- Verlag, New York, 1986. (Cited on page xvii).
[33] P.A.M., Dirac.The Principles of Quantum Mechanics. Oxford, at the Clarendon Press, 1947. 3rd ed.
[34] M., Duflo and C.C., Moore.On the regular representation of a nonunimodular locally compact group. J. Funct. Anal., 21(2):209–243, 1976. (Cited on page 114).
[35] A., Erdélyi, W., Magnus, F., Oberhettinger, and F.G., Tricomi.Higher Transcendental Functions, volume 2. McGraw Hill, New York, 1953. (Cited on page 264).
[36] P., Feinsilver and J., Kocik.Krawtchouk matrices from classical and quantum random walks. In Algebraic methods in statistics and probability. AMS special session on algebraic methods in statistics, Univ. of Notre Dame, IN, USA, April 8–9, 2000, pages 83–96. Providence, RI: AMS, American Mathematical Society, 2001. (Cited on page 72).
[37] P., Feinsilver and R., Schott.Krawtchouk polynomials and finite probability theory. In Probability Measures on groups X. Proceedings of the Tenth Oberwolfach conference, held November 4-10, 1990 in Oberwolfach, Germany, pages 129–135. New York, NY: Plenum Publishing Corporation, 1991. (Cited on page 72).
[38] P., Feinsilver and R., Schott.Algebraic structures and operator calculus, Vol. I: Representations and Probability Theory, volume 241 of Mathematics and Its Applications. Kluwer Academic Publishers Group, Dordrecht, 1993. (Cited on pages xviii, 92, 94, and 95).
[39] P., Feinsilver and R., Schott.Algebraic structures and operator calculus. Vol. II, volume 292 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1994. Special functions and computer science. (Cited on page xv).
[40] P., Feinsilver and R., Schott.Algebraic structures and operator calculus. Vol. III, volume 347 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1996. Representations of Lie groups. (Cited on page 99).
[41] R.P., Feynman, R., Leighton, and M., Sands.The Feynman Lectures on Physics. Vols. 1-3. Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964- 1966. www.feynmanlectures.info (Cited on pages 70 and 72).
[42] U., Franz. Classical Markov processes from quantum Lévy processes. Inf. Dim. Anal., Quantum Prob., and Rel. Topics, 2(1):105–129, 1999. (Cited on pages xvii and 132).
[43] U., Franz, R., Léandre, and R., Schott.Malliavin calculus for quantum stochastic processes. C. R. Acad. Sci. Paris Sér. I Math., 328(11):1061–1066, 1999. (Cited on page xviii).
[44] U., Franz, R., Léandre, and R., Schott.Malliavin calculus and Skorohod integration for quantum stochastic processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 4(1):11–38, 2001. (Cited on page xviii).
[45] U., Franz and R., Schott.Stochastic Processes and Operator Calculus on Quantum Groups. Kluwer Academic Publishers G roup, Dordrecht, 1999. (Cited on pages 139 and 147).
[46] U., Franz and N., Privault.Quasi-invariance formulas for components of quantum Lévy processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 7(1):131– 145, 2004. (Cited on page 16).
[47] C.W., Gardiner and P., Zoller.Quantum Noise. Springer Series in Synergetics. Springer-Verlag, Berlin, second edition, 2000. A handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics. (Cited on pages xvii and 131).
[48] C., Gerry and P., Knight.Introductory Quantum Optics. Cambridge University Press, Cambridge, 2004. (Cited on page 128).
[49] L., Gross.Abstract Wiener spaces. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, 1967. Univ. of California Press. (Cited on page 173).
[50] A., Guichardet.Symmetric Hilbert spaces and related topics, volume 261 of Lecture Notes in Mathematics. Springer Verlag, Berlin, Heidelberg, New York, 1972. (Cited on page 88).
[51] A., Guichardet.Cohomologie des groupes topologiques et des algèbres de Lie., volume 2 of Textes Mathematiques. CEDIC/Fernand Nathan, Paris, 1980. (Cited on pages 147 and 148).
[52] T., Hida.Brownian Motion. Springer Verlag, Berlin 1981. (Cited on page 158).
[53] A.S., Holevo.Statistical structure of quantum theory, volume 67 of Lecture Notes in Physics. Monographs. Springer-Verlag, Berlin, 2001. (Cited on pages xvii and 131).
[54] R.L., Hudson and K.R., Parthasarathy.Quantum Itô's formula and stochastic evolutions. Comm. Math. Phys., 93(3):301–323, 1984. (Cited on pages 190, 212, 214, and 215).
[55] C.J., Isham.Lectures on quantum theory. Imperial College Press, London, 1995. Mathematical and structural foundations. (Cited on page xviii).
[56] Y., Ishikawa.Stochastic Calculus of Variations for Jump Processes. de Gruyter, Berlin, 2013. (Cited on page xviii).
[57] L., Isserlis.On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika, 12(1-2): 134–139, 1918. (Cited on page 240).
[58] S., Janson.Gaussian Hilbert spaces, volume 129 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1997. (Cited on pages 155 and 211).
[59] U.C., Ji and N., Obata.Annihilation-derivative, creation-derivative and representation of quantum martingales. Commun. Math. Phys., 286(2):751–775, 2009. (Cited on page 216).
[60] U.C., Ji and N., Obata.Calculating normal-ordered forms in Fock space by quantum white noise derivatives. Interdiscip. Inf. Sci., 19(2):201–211, 2013. (Cited on page 216).
[61] J.R., Johansson, P.D., Nation, and F., Nori.Qutip: An open-source python framework for the dynamics of open quantum systems. Computer Physics Communications, 183(8):1760–1772, 2012. (Cited on page 113).
[62] J.R., Johansson, P.D., Nation, and F., Nori. Qutip 2: A python framework for the dynamics of open quantum systems. Computer Physics Communications, 184(4):1234–1240, 2013. (Cited on page 113).
[63] R., Koekoek and R.F., Swarttouw. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Delft University of Technology, Report 98–17, 1998. (Cited on pages 40, 41, 237, and 238).
[64] A., Korzeniowski and D., Stroock. An example in the theory of hypercontractive semigroups. Proc. Amer. Math. Soc., 94:87–90, 1985. (Cited on pages 18 and 26).
[65] P.S. de, Laplace. Théorie Analytique des Probabilités. V. Courcier, Imprimeur, 57 Quai des Augustins, Paris, 1814.
[66] M., Ledoux.Concentration of measure and logarithmic Sobolev inequalities. In Séminaire de Probabilités XXXIII, volume 1709 of Lecture Notes in Math., pages 120–216. Springer, Berlin, 1999. (Cited on page 18).
[67] V.P., Leonov and A.N., Shiryaev. On a method of calculation of semi-invariants. Theory Probab. Appl., 4:319–329, 1959. (Cited on pages 239 and 240).
[68] J.M., Lindsay.Quantum and non-causal stochastic calculus. Probab. Theory Related Fields, 97:65–80, 1993. (Cited on pages 88, 190, 212, and 213).
[69] J.M., Lindsay. Integral-sum kernel operators. In Quantum Probability Communications (Grenoble, 1998), volume XII, page 121. World Scientific, Singapore, 2003. (Cited on page 88).
[70] J.M., Lindsay.Quantum stochastic analysis – an introduction. In Quantum independent increment processes. I, volume 1865 of Lecture Notes in Math., pages 181–271. Springer, Berlin, 2005. (Cited on page 88).
[71] E., Lukacs. Applications of Faà di Bruno's formula in mathematical statistics. Am. Math. Mon., 62:340–348, 1955. (Cited on pages 239 and 240).
[72] E., Lukacs.Characteristic Functions. Hafner Publishing Co., New York, 1970. Second edition, revised and enlarged. (Cited on pages 239 and 240).
[73] H., Maassen.Quantum markov processes on fock space described by integral kernels. In Quantum probability and applications II (Heidelberg 1984), volume 1136 of Lecture Notes in Math., pages 361–374. Springer, Berlin, 1985. (Cited on page 88).
[74] T., Mai, R., Speicher, and M., Weber. Absence of algebraic relations and of zero divisors under the assumption of full non-microstates free entropy dimension. Preprint arXiv:1502.06357, 2015. (Cited on page 216).
[75] P., Malliavin. Stochastic calculus of variations and hypoelliptic operators. In Intern. Symp. SDE. Kyoto, pages 195–253, Tokyo, 1976. Kinokumiya. (Cited on pages xiii, xvii, and 173).
[76] P., Malliavin.Stochastic analysis, volume 313 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1997. (Cited on page 155).
[77] P., Malliavin.Stochastic analysis, volume 313 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1997. (Cited on page 169).
[78] S., Mazur and W., Orlicz.Grundlegende Eigenschaften der polynomischen Operationen. I. Stud. Math., 5:50–68, 1934. (Cited on page 259).
[79] P.A., Meyer.Quantum probability for probabilists, volume 1538 of Lecture Notes in Math. Springer-Verlag, Berlin, 2nd edition, 1995. (Cited on pages 7, 88, 135, 147, 186, and 211).
[80] P.A., Meyer.Quantum probability seen by a classical probabilist. In Probability towards 2000 (New York, 1995), volume 128 of Lecture Notes in Statist., pages 235–248. Springer, New York, 1998. (Cited on page xvi).
[81] A., Nica and R., Speicher. Lectures on the combinatorics of free probability, volume 335 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2006. (Cited on page 88).
[82] M.A., Nielsen and I.L., Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2000. (Cited on page 70).
[83] D., Nualart. Analysis on Wiener space and anticipating stochastic calculus. In Ecole d’été de Probabilités de Saint-Flour XXV, volume 1690 of Lecture Notes in Mathematics, pages 123–227. Springer-Verlag, Berlin, 1998. (Cited on page 155).
[84] D., Nualart. The Malliavin calculus and related topics. Probability and its Applications. Springer-Verlag, Berlin, second edition, 2006. (Cited on pages xvii, 155, and 173).
[85] H., Oehlmann. Analyse temps-fréquence de signaux vibratoires de boîtes de vitesses. PhD thesis, Université Henri Poincaré Nancy I, 1996. (Cited on page 130).
[86] H., Osswald. Malliavin calculus for Lévy processes and infinite-dimensional Brownian motion, volume 191 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2012. (Cited on pages xviii and 169).
[87] K.R., Parthasarathy. An Introduction to Quantum Stochastic Calculus. Birkäuser, 1992. (Cited on pages xv, 7, 47, 48, 82, 86, 88, 135, 186, 208, 212, 214, and 225).
[88] K.R., Parthasarathy. Lectures on quantum computation, quantum errorcorrecting codes and information theory. Tata Institute of Fundamental Research, Mumbai, 2003. Notes by Amitava Bhattacharyya. (Cited on page 72).
[89] G.|Peccati and M., Taqqu.Wiener Chaos: Moments, Cumulants and Diagrams: A survey with Computer Implementation. Bocconi and Springer Series. Springer, Milan, 2011. (Cited on page 239).
[90] J., Pitman. Combinatorial stochastic processes, volume 1875 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2006. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. (Cited on page 239).
[91] N., Privault. Inégalités de Meyer sur l'espace de Poisson. C. R. Acad. Sci. Paris Sér. I Math., 318:559–562, 1994. (Cited on page 18).
[92] N., Privault.A transfer principle from Wiener to Poisson space and applications. J. Funct. Anal., 132:335–360, 1995. (Cited on page 26).
[93] N., Privault.A different quantum stochastic calculus for the Poisson process. Probab. Theory Related Fields, 105:255–278, 1996. (Cited on pages 16 and 265).
[94] N., Privault.Girsanov theorem for anticipative shifts on Poisson space. Probab. Theory Related Fields, 104:61–76, 1996. (Cited on page 173).
[95] N., Privault.Une nouvelle représentation non-commutative du mouvement brownien et du processus de Poisson. C. R. Acad. Sci. Paris Sér. I Math., 322:959–964, 1996. (Cited on page 16).
[96] N., Privault. Absolute continuity in infinite dimensions and anticipating stochastic calculus. Potential Analysis, 8(4):325–343, 1998. (Cited on page 173).
[97] N., Privault. Splitting of Poisson noise and Lévy processes on real Lie algebras. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5(1):21–40, 2002. (Cited on pages 16 and 265).
[98] N., Privault.Stochastic analysis in discrete and continuous settings with normal martingales, volume 1982 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009. (Cited on pages 149, 155, and 174).
[99] N., Privault.Generalized Bell polynomials and the combinatorics of Poisson central moments. Electron. J. Combin., 18(1):Research Paper 54, 10, 2011. (Cited on page 241).
[100] N., Privault and W., Schoutens. Discrete chaotic calculus and covariance identities. Stochastics Stochastics Rep., 72(3-4):289–315, 2002. (Cited on page 72).
[101] L., Pukanszky.Leçon sur les repréntations des groupes. Dunod, Paris, 1967. (Cited on page 142).
[102] L., Pukanszky. Unitary representations of solvable Lie groups. Ann. scient. Éc. Norm. Sup., 4:457–608, 1971. (Cited on page 142).
[103] R., Ramer. On nonlinear transformations of Gaussian measures. J. Funct. Anal., 15:166–187, 1974. (Cited on page 173).
[104] S., Sakai.C*-Algebras and W*-Algebras. Springer-Verlag, New York-Heidelberg, 1971. (Cited on page 179).
[105] M., Schürmann.The Azéma martingales as components of quantum independent increment processes. In J., Azéma, P.A., Meyer, and M., Yor, editors, Séminaire de Probabilités XXV, volume 1485 of Lecture Notes in Math. Springer-Verlag, Berlin, 1991. (Cited on pages xvii and 132).
[106] M., Schürmann.White Noise on Bialgebras. Springer-Verlag, Berlin, 1993. (Cited on pages xvii, 131, 134, 136, 147, and 179).
[107] I., higekawa. Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ., 20(2):263–289, 1980. (Cited on page 173).
[108] K.B., Sinha and D., Goswami. Quantum stochastic processes and noncommutative geometry, volume 169 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2007. (Cited on page xviii).
[109] R. F., Streater. Classical and quantum probability. J. Math. Phys., 41(6):3556– 3603, 2000. (Cited on pages xvii and 147).
[110] T.N., Thiele. On semi invariants in the theory of observations. Kjöbenhavn Overs., pages 135–141, 1899. (Cited on page 239).
[111] N., Tsilevich, A.M., Vershik, and M., Yor. Distinguished properties of the gamma process and related topics. math.PR/0005287, 2000. (Cited on pages xvii, 180, and 189).
[112] N., Tsilevich, A.M., Vershik, and M., Yor. An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process. J. Funct. Anal., 185(1):274–296, 2001. (Cited on pages xvii, 180, and 189).
[113] A.S., ü stünel. An introduction to analysis on Wiener space, volume 1610 of Lecture Notes in Mathematics. Springer Verlag, Berlin, 1995. (Cited on page 155).
[114] A.M., Vershik, I.M., Gelfand, and M.I., Graev. A commutative model of the group of currents SL(2,R)X connected with a unipotent subgroup. Funct. Anal. Appl., 17(2):137–139, 1983. (Cited on pages xvii and 189).
[115] N.J., Vilenkin and A.U., Klimyk. Representation of Lie groups and special functions. Vol. 1, volume 72 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991. (Cited on page 99).
[116] N.J., Vilenkin and A.U., Klimyk. Representation of Lie groups and special functions. Vol. 3, volume 75 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1992. (Cited on page 99).
[117] N.J., Vilenkin and A.U., Klimyk. Representation of Lie groups and special functions. Vol. 2, volume 74 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1993. (Cited on page 99).
[118] N.J., Vilenkin and A.U., Klimyk.Representation of Lie groups and special functions, volume 316 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1995. (Cited on page 99).
[119] J., Ville. Théorie et applications de la notion de signal analytique. Câbles et Transmission, 2:61–74, 1948. (Cited on page 130).
[120] D., Voiculescu. Lectures on free probability theory. In Lectures on probability theory and statistics (Saint-Flour, 1998), volume 1738 of Lecture Notes in Math., pages 279–349. Berlin: Springer, 2000. (Cited on page 88).
[121] D., Voiculescu, K., Dykema, and A., Nica. Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups, volume 1 of CRM Monograph Series. American Mathematical Society, Providence, RI, 1992. (Cited on page 88).
[122] W. von, Waldenfels. Itô solution of the linear quantum stochastic differential equation describing light emission and absorption. In Quantum probability and applications to the quantum theory of irreversible processes, Proc. int. Workshop, Villa Mondragone/Italy 1982, volume 1055 of Lecture Notes in Math., pages 384–411. Springer-Verlag, Berlin, 1984. (Cited on pages xvii and 131).
[123] W. von, Waldenfels. A measure theoretical approach to quantum stochastic processes, volume 878 of Lecture Notes in Physics. Monographs. Springer- Verlag, Berlin, 2014. (Cited on page xviii).
[124] E.P., Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749–759, 1932. (Cited on page xvi).
[125] M.W., Wong. Weyl Transforms. Universitext. Springer-Verlag, Berlin, 1998. (Cited on page 108).
[126] L.M., Wu. L1 and modified logarithmic Sobolev inequalities and deviation inequalities for Poisson point processes. Preprint, 1998. (Cited on page 167).

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