- We continue the investigation of the Lvy processes on a q-deformed full Fock space started in . First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property for it in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras.