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\(q\)-Gaussian processes: Non-commutative and classical aspects. (English) Zbl 0873.60087
Summary: We examine, for \(-1<q<1\), \(q\)-Gaussian processes, i.e. families of operators (non-commutative random variables) \(X_t=a_t+a_t^*\) – where the \(a_t\) fulfill the \(q\)-commutation relations \(a_sa_t^*-qa_t^*a_s=c(s,t)\cdot \mathbf{1}\) for some covariance function \(c(\cdot,\cdot)\) – equipped with the vacuum expectation state. We show that there is a \(q\)-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on \(q\)-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of \(q\)-Gaussian processes possesses a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of U. Frisch and K. Bourret [J. Math. Phys. 11, 364-390 (1970; Zbl 0187.25902)].

MSC:
60K40 Other physical applications of random processes
60G15 Gaussian processes
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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