Abstract. We examine, for −1 < q < 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) Xt = at + a ∗ t – where the at fulfill the q-commutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – 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].

.4> F(H) := C\Omega \Phi M n1 H\Omega n ; where\Omega is a distinguished unit vector, called vacuum. 2) The vacuum expectation is the state A 7! h\Omega ; A\Omega i: 1 2 ROLAND SPEICHER 3) For each f 2 H we define the (left) annihilation operator l(f) and the (left) creation operator l (f) by l(f)\Omega = 0 l(f)f 1\Omega \Delta \Delta \Delta\Omega f n = hf; f 1 if 2\Omega \Delta \Delta \Delta\Omega f n and l (f)f<F