-Gaussian processes: non-commutative and classical aspects
arXiv:funct-an/9604010 · doi:10.1007/s002200050084
Abstract
We examine, for , -Gaussian processes, i.e. families of operators (non-commutative random variables) -- where the fulfill the -commutation relations $a_sa_t^*-qa_t^*a_s=c(s,t)\cdot \id$ for some covariance function -- equipped with the vacuum expectation state. We show that there is a -analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on -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 -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 \cite{FB}.
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