Communications in Mathematical Physics

, Volume 137, Issue 3, pp 519–531 | Cite as

An example of a generalized Brownian motion

  • Marek Bożejko
  • Roland Speicher


We present an example of a generalized Brownian motion. It is given by creation and annihilation operators on a “twisted” Fock space ofL2(ℝ). These operators fulfill (for a fixed −1≦μ≦1) the relationsc(f)c*(g)−μc*(g)c(f)=〈f,g〉1 (f, gL2(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 =v2).


Neural Network Gaussian Distribution Statistical Physic Complex System Brownian Motion 
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© Springer-Verlag 1991

Authors and Affiliations

  • Marek Bożejko
    • 1
  • Roland Speicher
    • 2
  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland
  2. 2.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergFederal Republic of Germany

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