## q-Gaussian Processes: Non-Commutative And Classical Aspects (1995)

Citations: | 64 - 2 self |

### BibTeX

@MISC{Bozejko95q-gaussianprocesses:,

author = {Marek Bozejko and Burkhard Kümmerer and Roland Speicher},

title = {q-Gaussian Processes: Non-Commutative And Classical Aspects},

year = {1995}

}

### OpenURL

### 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 ...