It is widely known that the classical probability theory initiated by Kolmogorov and its quantum (or noncommutative) counterpart pioneered by von Neumann were both created at about the same time. However, the subsequent developments of the latter have trailed far behind the former. This is perhaps because development of a theory of quantum stochastics requires an unusually large number of tools from operator theory and perhaps also because the probabilistic and analytical tools for understanding sample path behaviors of quantum stochastic processes have yet to be developed. This monograph is intended to provide the interested readers with a systematic and yet introductory treatment of a theory of quantum Markov processes that is in parallel to its commutative counterpart, namely, the well-known classical theory of probability and Markov processes.

This monograph can be used as an introduction and/or as a research reference for researchers and advanced graduate students who have been exposed to the theory of classical (or commutative) probability and Markov processes and have a special interest in their non-commutative counterparts. This monograph is intended to be as self-contained as possible by providing necessary review material and the proofs for almost all of the lemmas, propositions, and theorems contained herein. Some knowledge in real analysis, functional analysis, and stochastic processes will be helpful. However, no background material is assumed beyond knowledge of the basic theory of Hilbert spaces, bounded linear operators, and classical Markov processes.

This monograph is largely based on a current account of relevant research results contributed by many researchers on quantum stochastic calculus, quantum dynamical or Markov semigroups, Markov dilations, quantum Markov processes, and large time asymptotic behaviors such as those of the invariant states, recurrence and transience, ergodicity, and stability of quantum Markov semigroups/processes. The bibliography certainly not exhaustive and is likely to have omitted works by other researchers.