  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}.
