Recurrence and Transience of Quantum Markov Semigroups

Abstract

This article introduces concepts and surveys recent results on recurrence and transience of general quantum Markov semigroups (QMS) of bounded linear maps acting on a C*- or von Neumann algebra . In particular, the concept of potentials for classical Markov semigroups/processes is extended to its noncommutative counterpart. The characterization of recurrent and transient quantum Markov semigroups and classification of irreducible quantum Markov semigroups are established in terms of the potential of some subharmonic projection for the QMS. This introductory and survey work can be treated as a continuation of the closely related paper by Chang [12], which dealt with the invariance, mean ergodicity and ergodicity of QMS. Since it is intended as an introduction to large time asymptotic behavior of quantum Markov semigroups, this article is made self-contained by reviewing relevant concepts and results in quantum probability space, quantum states, and quantum Markov semigroups that are necessary for the subsequent developments and readability for nonexperts in this research areas.