$q$-Gaussian processes: non-commutative and classical aspects

TL;DR

This paper explores $q$-Gaussian processes, revealing their structure through a $q$-analogue of second quantization, and demonstrates that many possess a non-commutative Markov property with classical counterparts.

Contribution

It introduces a $q$-analogue of the Gaussian functor for $q$-Gaussian processes and shows these processes can have classical versions, answering an open question.

Findings

Existence of a $q$-analogue of the Gaussian functor

Many $q$-Gaussian processes have a non-commutative Markov property

These processes can be related to classical stochastic processes

Abstract

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 $a_sa_t^*-qa_t^*a_s=c(s,t)\cdot \id$ for some covariance function $c(\cdot,\cdot)$ -- 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 \cite{FB}.