q-Gaussian Processes: Non-commutative and Classical Aspects
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We examine, for −1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) \(\)– where the at fulfill the q-commutation relations \(\) for some covariance function \(\)– 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 Frisch and Bourret [FB].
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