QUANTUM FRACTIONAL ORNSTEIN–UHLENBECK SEMIGROUPS AND ASSOCIATED POTENTIALS

Abstract

Using an infinite-dimensional nuclear space, we introduce the quantum fractional number operator (QFNO) and the associated quantum fractional Ornstein–Uhlenbeck (O–U) semigroups. Then, we solve the Cauchy problems associated with the QFNO and show that its solutions can be expressed in terms of the aforementioned semigroups. Besides, we prove that the quantum fractional O–U semigroups satisfy the Feller property. Finally, using an adequate definition of the quantum fractional potentials, we give the solutions of the quantum fractional Poisson equations.