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1 results

  • Nonlinear Markov chains with finite state space: invariant distributions and long-term behaviour

  • Berenice Anne Neumann
  • Journal:
    Journal of Applied Probability / Volume 60 / Issue 1 / March 2023
    Published online by Cambridge University Press:
    22 September 2022, pp. 30-44
    Print publication:
    March 2023
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  • Nonlinear Markov chains with finite state space were introduced by Kolokoltsov (Nonlinear Markov Processes and Kinetic Equations, 2010). The characteristic property of these processes is that the transition probabilities depend not only on the state, but also on the distribution of the process. Here we provide first results regarding their invariant distributions and long-term behaviour: we show that under a continuity assumption an invariant distribution exists and provide a sufficient criterion for the uniqueness of the invariant distribution. Moreover, we present examples of peculiar limit behaviour that cannot occur for classical linear Markov chains. Finally, we present for the case of small state spaces sufficient (and easy-to-verify) criteria for the ergodicity of the process.