Abstract

. We examine, for \Gamma1 ! 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 asa t \Gamma qa t as = c(s; t) \Delta 1 for some covariance function c(\Delta; \Delta) -- 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 possess 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]. Introduction What we are going to call q-Gaussian processes was essentially introduced in a remarkable paper by Frisch ...

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