Ergodicity of p−majorizing nonlinear Markov operators on the finite dimensional space

Abstract

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. The nonlinear Markov chain over a finite state space can be identified by a continuous mapping (the so-called 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. In this paper, we introduce a notion of Dobrushin's ergodicity coefficients for stochastic hypermatrices and provide a criterion for the contraction nonlinear Markov operator by means of Dobrushin's ergodicity coefficients. We also introduce a notion of p−majorizing nonlinear Markov operators associated with stochastic hypermatrices and provide a criterion for strong ergodicity of such kind of operator. We show that the p−majorizing nonlinear Markov operators associated with scrambling, Sarymsakov, and Wolfowitz stochastic hypermatrices are strongly ergodic. These classes of p−majorizing nonlinear Markov operators assure an existence of a residual set of strongly ergodic nonlinear Markov operators which are not contractions. Some supporting examples are also provided.