q-Gaussian processes: Non-commutative and classical aspects

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(s)a(t)* - qa(t)*a(s) = c(s, t) . 1 for some covariance function c(.,.) - 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].