Scrambling, Sarymsakov, and Wolfowitz p–stochastic Hypermatrices and Dobrushin's Ergodicity Coefficients

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. 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 talk, 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. This talk is based on the results published in the papers [1–4].

• M. Saburov. Ergodicity of nonlinear Markov operators on the finite dimensional space. Nonlinear Analysis, 143 (2016) 105–119.
  
• M. Saburov. Ergodicity of p–majorizing quadratic stochastic operators. Markov Processes and Related Fields, 24(1) (2018) 131–150.
  
• M. Saburov. Ergodicity of p–majorizing nonlinear Markov operators on the finite dimensional space. Linear Algebra and its Applications, 578 (2019) 53–74.
  
• M. Saburov. Ergodicity of non-homogeneous p–majorizing quadratic stochastic operators. Positivity, 24(5) (2020) 1191–1209.