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An example of a generalized Brownian motion. (English) Zbl 0722.60033
Summary: We present an example of a generalized Brownian motion. It is given by creation and annihilation operators on a “twisted” Fock space of \(L^ 2({\mathbb{R}})\). These operators fulfill (for a fixed -1\(\leq \mu \leq 1)\) the relations \(c(f)c^*(g)-\mu c^*(g)c(f)=<f,g>1\) \((f,g\in L^ 2({\mathbb{R}}))\). 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 \(S_{\nu}U(n)\) of Woronowicz (with \(\mu =\nu^ 2)\).

60G20 Generalized stochastic processes
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
60J65 Brownian motion
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