Quadratic Stochastic Processes of Type $\boldsymbol{(\sigma|\mu)}$

#### B. J. Mamurov, U.A. Rozikov, S. S. Xudayarov

2020, v.26, Issue 5, 915-933

ABSTRACT

We construct quadratic stochastic processes (QSP) (also known as Markov processes of cubic matrices)
in continuous and discrete times. These are dynamical systems given by (a fixed type,
called $\sigma$) stochastic cubic matrices
satisfying an analogue of Kolmogorov\tire Chapman equation (KCE) with respect to
a fixed multiplications (called $\mu$) between
cubic matrices. The existence of a stochastic (at each time)
solution to the KCE provides
the existence of a QSP called a QSP of type $(\sigma | \mu)$.

In this paper, our aim is to construct and study trajectories of QSPs
for specially chosen notions of stochastic cubic matrices
and a wide class of multiplications of such matrices (known as Maksimov's multiplications).

Keywords: quadratic stochastic dynamics; cubic matrix; Kolmogorov\tire Chapman equation; time