, Volume 185, Issue 1, pp 129-154

q-Gaussian Processes: Non-commutative and Classical Aspects

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Abstract:

We examine, for −1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) – where the a t 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].

Received: 25 July 1996 / Accepted: 17 September 1996