Quantum dynamical semigroups induced by Gaussian generalized Mehler semigroups

Based on white noise theory, we first discuss the Gaussian generalized Mehler semigroups (on white noise functionals), along with their fundamental properties: invariant measures, invariant white noise operators, and mean ergodicity. Then, utilizing canonical topological isomorphisms between the spaces of two-variable white noise functionals and the spaces of white noise operators, we formulate a quantum analogue of the Gaussian generalized Mehler semigroups and discuss their fundamental properties. By establishing the notion of the positivity of white noise operators, we introduce the concept of quantum dynamical (Markov) semigroups acting on the space of white noise operators. Finally, we discuss the Gaussianity of the quantum dynamical semigroups induced by the Gaussian generalized Mehler semigroups. The construction of the quantum dynamical semigroups leads to the general notion of quantum dynamical semigroups on generalized operators.