Ergodic properties of discrete quadratic stochastic processes defined on von Neumann algebras (Q2710712)

The investigation of many physical systems can be reduced to that of Markov processes connected with these systems. However, there are systems that are not described by Markov processes. Some problems in the natural sciences involve quantum quadratic stochastic operators. It concerns, for example, statistical mechanics, population genetics, etc. NEWLINENEWLINENEWLINEIn this paper the quantum quadratic stochastic processes are studied. Necessary and sufficient conditions for the validity of the ergodic principle are obtained. From the physical point of view this means that for sufficiently large values of time the system described by this process does not depend on the initial state of the system. Some ergodic properties of these processes were considered by the same authors earlier [Uzb. Mat. Zh. 1997, No. 3, 8-20 (1997; Zbl 0930.60023)]. The conditions under which these processes are regular are established. Relations between quantum quadratic stochastic processes and non-commutative Markov processes are found. The concept of quadratic stochastic operator was introduced by \textit{S. N. Bernstein} [Charkov Ann. Sci. 1, 83-115 (1924; JFM 50.0342.02)].