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 Vladikavkaz. Mat. Zh., 2017, Volume 19, Number 4, Pages 70–75 (Mi vmj634)

A note on surjective polynomial operators

M. Saburov

Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box, Kuantan, Pahang, 25200, Malaysia

Abstract: A linear Markov chain is a discrete time stochastic process whose transitions depend only on the current state of the process. A nonlinear Markov chain is a discrete time stochastic process whose transitions may depend on both the current state and the current distribution of the process. These processes arise naturally in the study of the limit behavior of a large number of weakly interacting Markov processes. The nonlinear Markov processes were introduced by McKean and have been extensively studied in the context of nonlinear Chapman–Kolmogorov equations as well as nonlinear Fokker–Planck equations. The nonlinear Markov chain over a finite state space can be identified by a continuous mapping (a nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex) of the finite state space and by a family of transition matrices depending on occupation probability distributions of states. Particularly, a linear Markov operator is a linear operator associated with a square stochastic matrix. It is well-known that a linear Markov operator is a surjection of the simplex if and only if it is a bijection. The similar problem was open for a nonlinear Markov operator associated with a stochastic hyper-matrix. We solve it in this paper. Namely, we show that a nonlinear Markov operator associated with a stochastic hyper-matrix is a surjection of the simplex if and only if it is a permutation of the Lotka–Volterra operator.

Key words: stochastic hyper-matrix, polynomial operator, Lotka–Volterra operator.

 Funding Agency Grant Number Ministry of Higher Education, Malaysia FRGS14-141-0382 This work has been partially supported by the MOHE grant FRGS14-141-0382.

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UDC: 517.9
MSC: 47H60, 47N10
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Citation: M. Saburov, “A note on surjective polynomial operators”, Vladikavkaz. Mat. Zh., 19:4 (2017), 70–75

Citation in format AMSBIB
\Bibitem{Sab17} \by M.~Saburov \paper A note on surjective polynomial operators \jour Vladikavkaz. Mat. Zh. \yr 2017 \vol 19 \issue 4 \pages 70--75 \mathnet{http://mi.mathnet.ru/vmj634}