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 of L2(R). These operators fulfill (for a fixed -1 less-than-or-equal-to-mu-less-than-or-equal-to 1) the relations c(f)c*(g)-mu-c*(g)c(f) = <f,g>1 (f,g-epsilon-L2(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 an n-dimensional Brownian motion which is invariant under the quantum group S(v)U(n) of Woronowicz (with-mu = v2).