Skip to main content

An example of a generalized Brownian motion

Abstract

We present an example of a generalized Brownian motion. It is given by creation and annihilation operators on a “twisted” Fock space ofL 2(ℝ). These operators fulfill (for a fixed −1≦μ≦1) the relationsc(f)c * (g)−μc * (g)c(f)=〈f,g〉1 (f, gL 2(ℝ)). We show that the distribution of these operators with respect to the vacuum expectation is a generalized Gaussian distribution, in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula. We also indicate that our Brownian motion is one component of ann-dimensional Brownian motion which is invariant under the quantum groupS ν U(n) of Woronowicz (withμ =v 2).

This is a preview of subscription content, access via your institution.

References

  • [ApH] Applebaum, D.B., Hudson, R.L.: Fermion Ito's formula and stochastic evolutions. Commun. Math. Phys.96, 473–496 (1984)

    Google Scholar 

  • [Avi] Avitzour, D.: Free products ofC *-algebras. Trans. Am. Math. Soc.271, 423–435 (1982)

    Google Scholar 

  • [BJS] Bożejko, M., Januszkiewics, T., Spatzier, R.J.: Infinite Coxeter groups do not have Kazhdan's property. J. Operator Theory,19, 63–68 (1988)

    Google Scholar 

  • [BSW] Barnett, C., Streater, R.F., Wilde, I.F.: The Ito-Clifford-integral: J. Funct. Anal.48, 172–212 (1982)

    Google Scholar 

  • [CoH] Cockroft, A.M., Hudson, R.L.: Quantum mechanical Wiener process. J. Multivariate Anal.7, 107–124 (1977)

    Google Scholar 

  • [CuH] Cushen, C.D., Hudson, R.L.: A quantum-mechanical central limit theorem. J. Appl. Prob.8, 454–469 (1971)

    Google Scholar 

  • [Cun] Cuntz, J.: SimpleC *-algebras generated by isometries. Commun. Math. Phys.57, 173–185 (1977)

    Google Scholar 

  • [Eva] Evans, D.E.: OnO n . Publ. RIMS, Kyoto Univ.16, 915–927 (1980)

    Google Scholar 

  • [GvW] Giri, N., von Waldenfels, W.: An algebraic version of the central limit theorem. Z. Wahrscheinlichkeitstheorie Verw. Gebiete42, 129–134 (1978)

    Google Scholar 

  • [Gre] Greenberg, O.W.:Q-mutators and violations of statistics. University of Maryland Preprint 91-034, 1990

  • [Heg] Hegerfeld, G.C.: Noncommutative analogs of probabilistic notions and results. J. Funct. Anal.64, 436–456 (1985)

    Google Scholar 

  • [HuP] Hudson, R.L., Parthasarathy, K.R.: Quantum Ito's formula and stochastic evolution. Commun. Math. Phys.93, 301–323 (1984)

    Google Scholar 

  • [Küm 1] Kümmerer, B.: Survey on a theory of non-commutative stationary Markov processes. In: Quantum probability and applications. III. Oberwolfach 1987, Lecture Notes in Mathematics, vol. 1303. Berlin, Heidelberg, New York: Springer 1988, pp. 154–182

    Google Scholar 

  • [Küm 2] Kümmerer, B.: Markov dilations and non-commutative Poisson processes. Preprint

  • [KPr] Kümmerer, B., Prin, J.: Generalized white noise and non-commutative stochastic integration. Preprint

  • [KSp] Kümmerer, B., Speicher, R.: Stochastic integration on the Cuntz-algebraO . Preprint

  • [Maa 1] Maassen, H.: Theoretical concepts in quantum probability; quantum Markov processes. Preprint, Nijmegen 1988

  • [Maa 2] Maassen, H.: Quantum Markov processes on Fock space described by integral kernels. In: Quantum probability and applications. II. Heidelberg 1984, Lecture Notes in Mathematics, vol. 1136, Berlin, Heidelberg, New York: Springer 1985, pp. 361–374

    Google Scholar 

  • [LiP] Lindsay, J.M., Parthasarathy, K.R.: Cohomology of power sets with applications in quantum probability. Commun. Math. Phys.124, 337–364 (1989)

    Google Scholar 

  • [PWo] Pusz, W., Woronowicz, S.L.: Twisted second quantization. Preprint

  • [Sch] Schur, I.: Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. J. Reine Angew. Math.140, 1–29 (1911)

    Google Scholar 

  • [Sma] Schürmann, M.: Quantum stochastic processes with independent and additive increments. Preprint, Heidelberg 1989

  • [Spe 1] Speicher, R.: A new example of “Independence” and “White noise”. Probab. Th. Rel. Fields84, 141–159 (1990)

    Google Scholar 

  • [Spe 2] Speicher, R.: Stochastic integration on the full Fock space with the help of a kernel calculus. To appear in Publ. RIMS27 (1991)

  • [Spe 3] Speicher, R.: In preparation

  • [Voi 1] Voiculescu, D.: Symmetries of some reduced free productC *-algebras. In: Operator algebras and their connection with topology and ergodic theory. Busteni, Romania, 1983, Lecture Notes in Mathematics, vol. 1132, Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  • [Voi 2] Voiculescu, D.: Addition of certain non-commuting random variables. J. Funct. Anal.66, 323–346 (1986)

    Google Scholar 

  • [vWa] von Waldenfels, W.: An algebraic central limit theorem in the anti-commuting case. Z. Wahrscheinlichkeitstheorie Verw. Gebiete42, 135–140 (1978)

    Google Scholar 

  • [Wor 1] Woronowicz, S.L.: TwistedSU(2) group. An example of a non-commutative differential calculus. Publ. RIMS23, 117–181 (1987)

    Google Scholar 

  • [Wor 2] Woronowicz, S.L.: Tannaka-Krein duality for compact matrix pseudogroups. TwistedSU(N) groups. Invent. Math.93, 35–76 (1988)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by A. Connes

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bożejko, M., Speicher, R. An example of a generalized Brownian motion. Commun.Math. Phys. 137, 519–531 (1991). https://doi.org/10.1007/BF02100275

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02100275

Keywords

  • Neural Network
  • Gaussian Distribution
  • Statistical Physic
  • Complex System
  • Brownian Motion