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, g ∈L 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).
Similar content being viewed by others
References
[ApH] Applebaum, D.B., Hudson, R.L.: Fermion Ito's formula and stochastic evolutions. Commun. Math. Phys.96, 473–496 (1984)
[Avi] Avitzour, D.: Free products ofC *-algebras. Trans. Am. Math. Soc.271, 423–435 (1982)
[BJS] Bożejko, M., Januszkiewics, T., Spatzier, R.J.: Infinite Coxeter groups do not have Kazhdan's property. J. Operator Theory,19, 63–68 (1988)
[BSW] Barnett, C., Streater, R.F., Wilde, I.F.: The Ito-Clifford-integral: J. Funct. Anal.48, 172–212 (1982)
[CoH] Cockroft, A.M., Hudson, R.L.: Quantum mechanical Wiener process. J. Multivariate Anal.7, 107–124 (1977)
[CuH] Cushen, C.D., Hudson, R.L.: A quantum-mechanical central limit theorem. J. Appl. Prob.8, 454–469 (1971)
[Cun] Cuntz, J.: SimpleC *-algebras generated by isometries. Commun. Math. Phys.57, 173–185 (1977)
[Eva] Evans, D.E.: OnO n . Publ. RIMS, Kyoto Univ.16, 915–927 (1980)
[GvW] Giri, N., von Waldenfels, W.: An algebraic version of the central limit theorem. Z. Wahrscheinlichkeitstheorie Verw. Gebiete42, 129–134 (1978)
[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)
[HuP] Hudson, R.L., Parthasarathy, K.R.: Quantum Ito's formula and stochastic evolution. Commun. Math. Phys.93, 301–323 (1984)
[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
[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
[LiP] Lindsay, J.M., Parthasarathy, K.R.: Cohomology of power sets with applications in quantum probability. Commun. Math. Phys.124, 337–364 (1989)
[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)
[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)
[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
[Voi 2] Voiculescu, D.: Addition of certain non-commuting random variables. J. Funct. Anal.66, 323–346 (1986)
[vWa] von Waldenfels, W.: An algebraic central limit theorem in the anti-commuting case. Z. Wahrscheinlichkeitstheorie Verw. Gebiete42, 135–140 (1978)
[Wor 1] Woronowicz, S.L.: TwistedSU(2) group. An example of a non-commutative differential calculus. Publ. RIMS23, 117–181 (1987)
[Wor 2] Woronowicz, S.L.: Tannaka-Krein duality for compact matrix pseudogroups. TwistedSU(N) groups. Invent. Math.93, 35–76 (1988)
Author information
Authors and Affiliations
Additional information
Communicated by A. Connes
Rights 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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02100275