Wick calculus for nonlinear Gaussian functionals

  • Yao-zhong Hu
  • Jia-an Yan


This paper surveys some results on Wick product and Wick renormalization. The framework is the abstract Wiener space. Some known results on Wick product and Wick renormalization in the white noise analysis framework are presented for classical random variables. Some conditions are described for random variables whose Wick product or whose renormalization are integrable random variables. Relevant results on multiple Wiener integrals, second quantization operator, Malliavin calculus and their relations with the Wick product and Wick renormalization are also briefly presented. A useful tool for Wick product is the S-transform which is also described without the introduction of generalized random variables.


Malliavin calculus multiple integral chaos decomposition Wick product Wick renormalization 

2000 MR Subject Classification

60G15 60H05 60H07 60H40 


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  1. [1]
    Budhiraja, A., Kallianpur, G. The generalized Hu-Meyer formula for random kernels. Appl. Math. Optim., 35(2): 177–202 (1997)zbMATHMathSciNetGoogle Scholar
  2. [2]
    Dellacherie, C., Meyer, P.A., Maisoneuve, B. Probabilits et Potentiel. Chapitres XVII–XXIV: Processus de Markov (fin), Complments de calcul stochastique, Hermann, Paris, 1992Google Scholar
  3. [3]
    Duncan, T.e E., Hu, Y., Pasik-Duncan, B. Stochastic calculus for fractional Brownian motion. I. Theory. SIAM J. Control Optim., 38(2): 582–612 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Glimm, J., Jaffe, A. Quantum physics. A functional integral point of view. Second edition. Springer-Verlag, New York, 1987Google Scholar
  5. [5]
    Hida, T., Ikeda, N. Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral. Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66). Vol. II: Contributions to Probability Theory, Part 1 pp. 117–143 Univ. California Press, California, 1967Google Scholar
  6. [6]
    Holden, H., Øksendal, B., Ubóe, J., Zhang, T. Stochastic partial differential equations. A modeling, white noise functional approach. Probability and its Applications. Birkhäuser Boston, 1996Google Scholar
  7. [7]
    Hu, Y. A unified approach to several inequalities for Gaussian and diffusion measures. Séminaire de Probabilités, XXXIV, 329–335. Lecture Notes in Math., 1729, Springer-Verlag, Berlin, 2000CrossRefGoogle Scholar
  8. [8]
    Hu, Y. Integral transformations and anticipative calculus for fractional Brownian motions. Mem. Amer. Math. Soc., 175, 2005Google Scholar
  9. [9]
    Hu, Y., Kallianpur, G. Exponential integrability and application to stochastic quantization. Appl. Math. Optim., 37(3): 295–353 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Hu, Y. and Meyer, P.A. Chaos de Wiener et intégrale de Feynman. Séminaire de Probabilités, XXII, 51–71, Lecture Notes in Math., 1321, Springer-Verlag, Berlin, 1988CrossRefGoogle Scholar
  11. [11]
    Hu, Y., Meyer, P.A. Sur les intgrales multiples de Stratonovitch. Séminaire de Probabilités, XXII, 72–81, Lecture Notes in Math., 1321, Springer-Verlag, Berlin, 1988CrossRefGoogle Scholar
  12. [12]
    Hu, Y., Meyer, P.A. On the approximation of multiple Stratonovich integrals. Stochastic processes, 141–147, Springer-Verlag, New York, 1993Google Scholar
  13. [13]
    Hu, Y. and Øksendal, B. Wick approximation of quasilinear stochastic differential equations. Stochastic analysis and related topics, V (Silivri, 1994), 203–231, Progr. Probab., 38, Birkhäuser Boston, 1996Google Scholar
  14. [14]
    Huang, Z.Y., Yan, J.A. Introduction to infinite dimensional stochastic analysis. Mathematics and its Applications, 502. Kluwer Academic Publishers, Dordrecht; Science Press, Beijing, 2000zbMATHGoogle Scholar
  15. [15]
    Johnson, G.W., Kallianpur, G. Homogeneous chaos, p-forms, scaling and the Feynman integral. Trans. Amer. Math. Soc., 340(2): 503–548 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Kallenberg, O. On an independence criterion for multiple Wiener integrals. Ann. Probab., 19(2): 483–485 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Kondratiev, Y.G., Streit, L.; Westerkamp, W., Yan, J.A. Generalized functions in infinite-dimensional analysis. Hiroshima Math. J., 28(2): 213–260 (1998)zbMATHMathSciNetGoogle Scholar
  18. [18]
    Kondratiev, Y.G., Leukert, P., Streit, L. Wick calculus in Gaussian analysis. Acta Appl. Math., 44(3): 269–294 (1996)zbMATHMathSciNetGoogle Scholar
  19. [19]
    Meyer, P.A. Transformations de Riesz pour les lois gaussiennes. Seminar on probability, XVIII, 179–193, Lecture Notes in Math., 1059, Springer, Berlin, 1984Google Scholar
  20. [20]
    Meyer, P.A. Quantum probability for probabilists. Lecture Notes in Mathematics, 1538. Springer-Verlag, Berlin, 1993zbMATHGoogle Scholar
  21. [21]
    Meyer, P.A., Yan, J.A. Distributions sur l’espace de Wiener (suite) d’aprés I. Kubo et Y. Yokoi. Séminaire de Probabilités, XXIII, 382–392, Lecture Notes in Math., 1372, Springer-Verlag, Berlin, 1989CrossRefGoogle Scholar
  22. [22]
    Pisier, G. Riesz transforms: a simpler analytic proof of P.-A. Meyer’s inequality. Séminaire de Probabilités, XXII, 485–501, Lecture Notes in Math., 1321, Springer-Verlag, Berlin, 1988CrossRefGoogle Scholar
  23. [23]
    Simon, B. The P(ϕ)2 Euclidean (quantum) field theory. Princeton Series in Physics. Princeton University Press, Princeton, 1974Google Scholar
  24. [24]
    Ustunel, A.S., Zakai, M. On independence and conditioning on Wiener space. Ann. Probab., 17(4): 1441–1453 (1989)CrossRefMathSciNetGoogle Scholar
  25. [25]
    Wick, G.C. The evaluation of the collision matrix. Physical Rev., 80(2): 268–272 (1950)zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    Yan, J.A. Notes on the Wiener semigroup and renormalization. Séminaire de Probabilités, XXV, 79–94, Lecture Notes in Math., 1485. Springer-Verlag, Berlin, 1991CrossRefGoogle Scholar
  27. [27]
    Yan, J.A. Products and transforms of white-noise functionals (in general setting). Appl. Math. Optim., 31(2): 137–153 (1995)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA
  2. 2.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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