In this talk, based on quantum white noise theory, we first discuss the Gaussian generalized Mehler semigroups, 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 semigroupsand discuss their fundamental properties.

  By establishing 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.