Generators of quantum one-dimensional diffusions

Quantum dynamical semigroups represent a noncommutative analogue of (sub)Markov semigroups in classical probability: while the latter are semigroups of maps in functional spaces, the former are semigroups of maps in operator algebras having certain properties of positivity and normalization. In this paper we describe quantum dynamical semigroups, which are the noncommutative analogues of classical diffusions on $\mathbf{R}$ and $\mathbf{R}_{+}$, and demonstrate various properties of the semigroup and its generator depending on the boundary condition. We also give a proof of a result describing the domain of the generator of “noncommutative diffusion on $\mathbf{R}_{+}$ with extinction at 0” and give an explicit example of the trace-class operator in this domain, which does not belong to the domain of closure of the initial operator.