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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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