Markov processes of cubic stochastic matrices: Quadratic stochastic processes

We define Markov processes of cubic stochastic (in a fixed sense) matrices which are also called quadratic stochastic process (QSPs). A QSP is a particular case of a continuous-time dynamical system whose states are stochastic cubic matrices satisfying an analogue of the Kolmogorov-Chapman equation (KCE). The existence of a stochastic solution to the KCE provides the existence of a QSP. In this talk we give several examples of QSPs for two specially chosen notions of stochastic cubic matrices and two multiplications of such matrices (known as Maksimov’s multiplications). We will show a wide class of QSPs and give some time-dependent behavior of such processes. We give an example with applications to the Biology, constructing a QSP which describes the time behavior (dynamics) of a population with the possibility of twin births.