Deformations of probability distributions on non-commutative probability spaces

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In this project, a deformed family of the classical Meixner distributions was considered as one of important examples for quantum deformations, and non-commutative random variables having these deformed Meixner distributions were constructed on the deformed Fock space by using the corresponding deformed annihilation and creation operators.We have also investigated the fundamental theories on quantum deformed entropy and Fisher information. Especially, in the case of free probability, we have given another proofs for several inequalities on the relative free entropy and the free Fisher information based on the free diffusion processes.