Non-Linear Extensions of Classical and Quantum Stochastic Calculus and Essentially Infinite Dimensional Analysis

  • Luigi Accardi
  • Yun-Gang Lu
  • Igor Volovich
Part of the Lecture Notes in Statistics book series (LNS, volume 128)


It is likely (at least for its proponent) that quantum probability, or more generally algebraic probability shall play for probability a role analogous to that played by algebraic geometry for geometry: many will complain against a loss of immediate intuition, but this is compensated for by an increase in power, the latter being measured by the capacity of solving old problems, not only inside probability theory, or at least of bringing non-trivial contributions to their advancement. The present, reasonably satisfactory, balance between developement of new techniques and problems effectively solved by these new tools should be preserved in order to prevent implosion into a self-substaining circle of problems and the main route to achieve this goal is the same as for classical probability, namely to keep a strong contact with advanced mathematical developement on one side and with real statistical data, wherever they come from, on the other.


White Noise Central Limit Theorem Quantum Probability Stochastic Calculus Hilbert Module 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Accardi, L., Bach, A. (1987) The harmonic oscillator as quantum central limit of Bernoulli processes. Accept by: Prob. Th. and Rel. Fields, Volterra Preprint.Google Scholar
  2. [2]
    Accardi, L., Bach, A. (1987) Central limits of squeezing operators, in: Quantum Probability and Applications IV Springer LNM N. 1396 7–19.Google Scholar
  3. [3]
    Accardi, L., Quaegebeur, J. (1988) Ito algebras of Gaussian quantum fields. Journ. Funct. Anal. 85 213–263.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Accardi L. (1990) An outline of quantum probability. Unpublished manuscript (a Russian version was accepted for Publication in Uspehi Matem. Nauk. The second half of the paper has appeared in Trudi Moscov. Matem. Obsh. 56 (1995) english translation in: Trans. Moscow Math. Soc. 56 (1995) 235–270.)MathSciNetGoogle Scholar
  5. [5]
    Accardi, L., Gibilisco, P., Volovich I.V. (1994) Yang-Mills gauge fields as harmonic functions for the Lévy Laplacian Russian Journal of Mathematical Physics 2 235–250.MathSciNetzbMATHGoogle Scholar
  6. [6]
    Accardi, L., Lu, Y.G., Volovich I. Non-Commutative (Quantum) Probability, Master Fields and Stochastic Bosonization Volterra preprint, CVV-198-94, hep-th/9412246.Google Scholar
  7. [7]
    Accardi, L. (1995) Yang-Mills equations and Levy laplacians. in: Dirichlet forms and stochastic processes, Eds. Ma Z.M., Rockner M., Yan JA., Walter de Gruyter 1–24.Google Scholar
  8. [8]
    Accardi, L. (1997) On the axioms of probability theory. Plenary talk given at the Annual Meeting of the Deutsche Mathematiker-Vereiningung. To appear in: Jaheresberichte der Deutsche Mathematiker-Vereiningung.Google Scholar
  9. [9]
    Accardi, L. (1996) Applications of Quantum Probability to Quantum Theory. Lectures delivered at Nagoya University (based on [AcLuVo97a]).Google Scholar
  10. [10]
    Accardi, L., Lu, Y.G., Obata, N. Towards a nonlinear extension of stochastic calculus.Google Scholar
  11. [11]
    Accardi, L., Lu, Y.G., Volovich, I. Quantum theory and its stochastic limit. Monograph in preparation.Google Scholar
  12. [12]
    Accardi, L., Lu, Y.G., Volovich, I. White noise approach to stochastic calculus and nonlinear Ito tables Submitted to: Nagoya Journal of Mathematics.Google Scholar
  13. [13]
    Accardi, L., Lu, Y.G., Volovich, I. (1997) Interacting Fock spaces and Hilbert module extensions of the Heisenberg commutation relations, to appear in: Preprint IIAS.Google Scholar
  14. [14]
    Arefeva, I., Volovich, I. (1996) The master field in QCD and q-deformed qauntum field theory. Nucl. Phys. B 462 600–613.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Choquet-Bruhat, Y. (1973) Distributions, Theorie et Problemes, Masson et C., Editeurs.zbMATHGoogle Scholar
  16. [16]
    Dellacherie, C., Meyer, P.A. (1975) Probabilites et potentiel. Hermann.zbMATHGoogle Scholar
  17. [17]
    Hida. T. (1975) Analysis of Brownian Functionals. Carleton Mathematical Lecture notes 13.Google Scholar
  18. [18]
    Hida, T. (1991) A Role of the Levy Laplacian on the Causal Calculus of Generalized White Noise Functionals. Preprint.Google Scholar
  19. [19]
    Hida, T., Obata, N., Saito, K. (1992) Infinite dimensional rotations and Laplacians in terms of white noise calculus. Nagoya Math. J. 128 65–93.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Hudson R.L., Parthasarathy K.R. (1994) Quantum Ito’s formula and stochastic evolutions, Commun. Math. Phys. 93 301–323.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Kuo, H.-H. (1996) White Noise Distribution Theory, CRC Press.zbMATHGoogle Scholar
  22. [22]
    Levy, P. (1951) Problemes concrets d’analyse fonctionnelle Gauthier Villars, Paris.zbMATHGoogle Scholar
  23. [23]
    Lu, Y.G. (1992) The Boson and Fermion Brownian Motion as Quantum Central limits of the Quantum Bernoulli Processes. Bollettino UMI, (7) 6-A, 245-273. Volterra preprint (1989).Google Scholar
  24. [24]
    Lu, Y.G., De Giosa, M. (1995) The free creation and annihilation operators as the central limit of quantum Bernoulli process. Preprint Dipartimento di Matematica Università di Bari 2.Google Scholar
  25. [25]
    Lu Y.G. The interacting Free Fock Space and the Deformed Wigner Law.Google Scholar
  26. [26]
    Meyer P.A. (1993) Quantum Probability for Probabilists, Lect. Notes in Math. Vol. 1538, Springer-Verlag.Google Scholar
  27. [27]
    Muraki, N. Noncommutative Brownian motion in monotone Fock space, to appear in Commun. Math. Phys.Google Scholar
  28. [28]
    Obata N. (1994) White Noise Calculus and Fock Space, Lect. Notes in Math. Vol. 1577, Springer-Verlag.Google Scholar
  29. [29]
    Obata N. (1995) Generalized quantum, stochastic processes on Fock space, Publ. RIMS 31 667–702.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    Ohya, M., Petz, D. (1993) Quantum entropy and its use, Springer, Texts and Monographs in Physics.zbMATHGoogle Scholar
  31. [31]
    Parthasarathy K.R. (1992) An Introduction to Quantum Stochastic Calculus, Birkhäuser.zbMATHGoogle Scholar
  32. [32]
    Skeide M. (1996) Hilbert modules in quantum electro dynamics and quantum probability. Volterra Preprint N. 257.Google Scholar
  33. [33]
    Voiculescu, D. (1991) Free noncommutative random variables, random matrices and the II 1 factors of free groups, in: Quantum Probability and related topics, World Scientific VI.Google Scholar
  34. [34]
    von Waldenfels, W., Giri, N. (1978) An Algebraic Version of the Central Limit Theorem. Z. Wahrscheinlichkeitstheorie verw. Gebiete 42, 129–134.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1998

Authors and Affiliations

  • Luigi Accardi
    • 1
    • 2
  • Yun-Gang Lu
    • 3
  • Igor Volovich
    • 4
  1. 1.Graduate School of PolymathematicsNagoya UniversityNagoyaJapan
  2. 2.Centro Matematico Vito VolterraUniversità di RomaRomaItaly
  3. 3.Dipartimento di MatematicaUniversità di BariBariItaly
  4. 4.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations