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

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

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


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


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