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Abstract This paper is the survey of some of our results related to q -deformations of the Fock spaces and related to q -convolutions for probability measures on the real line R R. The main idea is done by the combinatorics of moments of the measures and related q -cumulants of different types. The main and interesting q -convolutions are related to classical continuous (discrete) q -Hermite polynomial. Among them are classical (q=1 q = 1) convolutions, the case q=0 q = 0, gives the free and Boolean relations, and the new class of q -analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of q -convolutions. The main result is the construction of Brownian motion related to q -Discrete Hermite polynomial of type I.
Bożejko et al. (Sun,) studied this question.