  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 an 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. The most important ones are 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 is a random extension of the classical potential measure,
with a mean measure equal to infinity at every point of the state space.
