Coupling from the past for the null recurrent Markov chain

The Doeblin graph of a countable state space Markov chain describes the joint pathwise evolutions of the Markov dynamics starting from all possible initial conditions, with two paths coalescing when they reach the same point of the state space at the same time. Its bridge Doeblin subgraph only contains the paths starting from a tagged point of the state space at all possible times. In the irreducible, aperiodic, and positive recurrent case, the following results are known: the bridge Doeblin graph is an infinite tree that is unimodularizable. Moreover, it contains a single bi-infinite path which allows one to build a perfect sample of the stationary state of the Markov chain. The present paper is focused on the null recurrent case. It is shown that when assuming irreducibility and aperiodicity again, the bridge Doeblin graph is either a single infinite tree or a forest made of a countable collection of infinite trees. In the first case, the infinite tree in question has a single end, is not unimodularizable in general, but is always locally unimodular. These key properties are used to study the stationary regime of several measure-valued random dynamics on this bridge Doeblin Tree, which can be seen as pathwise extensions of classical distributional dynamics associated to the Markov chain. This includes the taboo random dynamics, which admits as steady state a random measure with mean measure equal to the invariant measure of the Markov chain, and the potential random dynamics which admits as steady state a locally finite random measure, with a mean measure equal to infinity at every point of the state space. The practical interest of these two random measures is discussed in the context of perfect sampling.