Nonlinear Markov Chains with Finite State Space: Invariant Distributions and Long-Term Behaviour

TL;DR

This paper investigates nonlinear Markov chains with finite states, establishing conditions for invariant distributions, analyzing their long-term behavior, and providing criteria for ergodicity, highlighting differences from classical linear Markov chains.

Contribution

It offers the first results on invariant distributions and long-term behavior of nonlinear Markov chains, including criteria for uniqueness and ergodicity, and examples of unique limit behaviors.

Findings

Existence of invariant distributions under continuity assumptions.

A sufficient criterion for uniqueness based on Brouwer degree.

Examples of peculiar limit behaviors not seen in linear Markov chains.

Abstract

Nonlinear Markov chains with finite state space have been introduced in Kolokoltsov (2010). The characteristic property of these processes is that the transition probabilities do not only depend on the state, but also on the distribution of the process. In this paper we provide first results regarding their invariant distributions and long-term behaviour. We will show that under a continuity assumption an invariant distribution exists. Moreover, we provide a sufficient criterion for the uniqueness of the invariant distribution that relies on the Brouwer degree. Thereafter, we will 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.