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DOI: 10.23671/VNC.2018.4.9169 Note on Surjective Polynomial Operators
Saburov M. A.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 4.
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 ChapmanKolmogorov equations as well as nonlinear FokkerPlanck 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 wellknown 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 hypermatrix. We solve it in this paper. Namely, we show that a nonlinear Markov operator associated with a stochastic hypermatrix is a surjection of the simplex if and only if it is a permutation of the LotkaVolterra operator.
Keywords: Stochastic hypermatrix, polynomial operator, LotkaVolterra operator
Language: English
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For citation: Saburov M. A Note on Surjective Polynomial Operators //
Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], 2017, vol.
19, no. 1, pp. 7075. DOI 10.23671/VNC.2018.4.9169 ← Contents of issue 
 

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