Quantum λ-potentials associated to quantum Ornstein–Uhlenbeck semigroups

Using the quantum Ornstein–Uhlenbeck (O–U) semigroups (introduced in Rguigui [21]) and based on nuclear infinite dimensional algebra of entire functions with a certain exponential growth condition with two variables, the quantum λ-potential and the generalised quantum (λ1,λ2)-potential appear naturally for λ,λ1,λ2∈(0,∞). We prove that the solution of Poisson equations associated with the suitable quantum number operators can be expressed in terms of these potentials. Using a useful criterion for the positivity of generalised operators, we demonstrate that the solutions of the Cauchy problems associated to the quantum number operators are positive operators if the initial condition is also positive. In this case, we show that these solutions, the quantum λ-potential and the generalised quantum (λ1,λ2)-potential have integral representations given by positive Borel measures. Based on a new notion of positivity of white noise operators, the aforementioned potentials are shown to be Markovian operators whenever λ∈[1,∞) and λ1λ2⩾λ1+λ2.