An example of a generalized Brownian motion


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).

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Communicated by A. Connes

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Bożejko, M., Speicher, R. An example of a generalized Brownian motion. Commun.Math. Phys. 137, 519–531 (1991).

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