On Marginal Processes of Quadratic Stochastic Processes

On cubic stochastic operators and processes

“…Next goal of the paper is to define cubic stochastic processes (CSP) and give differential equations for such processes. This investigations will be similar to works [8], [20,21], [24]- [26], where the authors introduced a continuous-time dynamical system as quadratic stochastic processes (QSP). The reader is referred to very recent book [19] for the theory of QSPs.…”

“…2. Note that if one defines a new process by Q(s, x, y, t, A) = E P (s, x, y, z, t, A)m s (dz) then one can check that the defined process is QSP (see [20,24]). We recall that the functions Q(s, x, y, t, A) denote the probability that under the interaction of the elements x and y at time s an event A comes into effect at time t. Since for physical, chemical and biological phenomena, a certain time is necessary for the realization of an interaction, it is taken the greatest such time to be equal to 1 (see the Boltzmann model [12] or the biological model [15]).…”

Prevalence problem in the set of quadratic stochastic operators acting on L1

“…One of the reasons for the growing interest in quadratic stochastic operators is their applicability to biological sciences, especially in the modelling of reproducing populations (Bartoszek and Pułka, 2013a;Ganikhodjaev et al, 2010Ganikhodjaev et al, , 2013Ganikhodjaev and Jamilov, 2012;Rudnicki and Zwoleński, 2015;Zwoleński, 2015) and describing multi-agent systems (Saburov and Saburov, 2014a,b). This interest has resulted in numerous recent works on general (Alexandrov, 2015;Mukhamedov and Supar, 2015;Mukhamedov et al, 2013) and ergodic (Ganikhodjaev and Zanin, 2012;Ganikhodjaev et al, 2006;Ganikhodzhaev and Zanin, 2004;Pułka, 2011;Saburov, 2007) properties of QSOs, specific subclasses of QSOs like Volterra (Mukhamedov, 2014;Mukhamedov and Saburov, 2010;Saburov, 2012Saburov, , 2013 or other (Ganikhodjaev and Hamzah, 2014). These new results build on a rich previous literature (e.g.…”

On stable b-bistochastic quadratic stochastic operators and associated non-homogenous Markov chains

“…Note that, this kind of construction was first considered in [4,20]. Certain properties of the associated Markov chains have been investigated in several paper such as [13,14,19] Let V : S n−1 → S n−1 be a q.s.o. defined by heredity coefficients {P ij,k } n i,j,k=1 and we denote x (m) j = (V (m) (x)) j , x ∈ S n−1 .…”

“…was first considered in [4,20]. Certain properties of the associated Markov chains have been investigated in several paper such as [13,14,19] According to the construction, the Markov measures depend on the initial state of x. Given q.s.o.…”