The question of recurrence and transience of branching Markov chains is more subtle than for ordinary Markov chains; they can be classified in transience, weak recurrence, and strong recurrence. We review criteria for transience and weak recurrence and give several new conditions for weak recurrence and strong recurrence. These conditions make a unified treatment of known and new examples possible and provide enough information to distinguish between weak and strong recurrence. This represents a step towards a general classification of branching Markov chains. In particular, we show that in homogeneous cases weak recurrence and strong recurrence coincide. Furthermore, we discuss the generalization of positive and null recurrence to branching Markov chains and show that branching random walks on $Z$ are either transient or positive recurrent.
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机译：与普通马尔可夫链相比，分支马尔可夫链的重复性和瞬时性问题更加微妙。它们可以分为短暂性，弱复发性和强复发性。我们回顾了短暂性和弱复发的标准，并给出了弱复发和强复发的几个新条件。这些条件使对已知和新示例的统一处理成为可能，并提供了足够的信息以区分弱复发和强复发。这代表了迈向分支马尔可夫链一般分类的一步。特别是，我们表明在同类情况下，弱复发和强复发是重合的。此外，我们讨论了正和零递归对分支Markov链的一般化，并显示了在$ Z $上的分支随机游走是瞬态的还是正性的。
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