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].

We derive the chaotic expansion of the product of n-th and first order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulas for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits properties relative to multiple stochastic integrals, polynomials and Wick products, that characterize the Wiener and Poisson processes. Key words: Az'ema martingales, multiple stochastic integrals, product formulas. Mathematics Subject Classification (1991): 60G44, 60H05, 81S25. 1 Introduction The Wiener-Ito and Poisson-Ito chaotic decompositions give an isometric isomorphism between the Fock space \Gamma(L 2 (IR + )) and the space of square-integrable functionals of the process. This somorphism is constructed by association of a symmetric function f n 2 L 2 (IR + ) ffin to its multiple stochastic integral. The isometry property comes ...

Monotone Lévy processes with additive increments are defined and studied. It is shown that these processes have natural Markov structure and their Markov transition semigroups are characterized using the monotone Lévy-Khintchine formula. 17 Monotone Lévy processes turn out to be related to classical Lévy processes via Attal’s “remarkable transformation. ” A monotone analogue of the family of exponential martingales associated to a classical Lévy process is also defined. 1.

ABSTRACT Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated to a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If H is any Hopf algebra, it may be realised in Lin(H) in such a way that ∆h = W(h ⊗ 1)W −1 for an operator W. This W is interpreted as the time evolution operator for the system at time t coupled quantum-mechanically to the system at time t + δ. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a CTP-type theorem. 1