论文标题
具有有限状态空间的非线性马尔可夫链：不变分布和长期行为
Nonlinear Markov Chains with Finite State Space: Invariant Distributions and Long-Term Behaviour
论文作者
论文摘要
Kolokoltsov（2010）中引入了具有有限状态空间的非线性马尔可夫链。这些过程的特征属性是过渡概率不仅取决于状态，还取决于该过程的分布。在本文中，我们提供了有关其不变分布和长期行为的首先结果。我们将表明，在连续性假设下存在不变分布。此外，我们为依赖Brouwer学位的不变分布的唯一性提供了足够的标准。此后，我们将介绍经典线性马尔可夫链无法发生的特殊极限行为的例子。最后，我们为小型状态空间的情况提供了足够的（易于验证）的标准，以实现该过程的刻薄性。
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.
