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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., 31 Pages, 8 figures

Let $\{X_n\} = \{X^{(N)}_n\}$ be a Markov chain in the probability measures $\mathbf{P}\lbrack 0, 1\rbrack$, equipped with a certain metric for the topology of weak convergence, and denote $\mathbf{E}_x(X_1) = \mathbf{f}_N(x)$. Define a projection $\xi = \pi_dx$ on $\mathbf{P}\lbrack 0, 1\rbrack$ by defining $\xi$ to be absolutely continuous with respect to Lebesgue measure with constant density $(d + 1)x(A)$ on each interval $A$ of an equipartition of $\lbrack 0, 1 \rbrack$ into $d + 1$ intervals, and let $\pi_d\mathbf{P}\lbrack 0, 1\rbrack \subset \mathbb{R}^d$ be the natural embedding. Assume $\pi_d\mathbf{f}_N(\xi) \underline{\sim} \xi + \beta_N h_d(\xi), h_d(\xi_{0, d}) = \mathbf{0}$ and $\operatorname{Cov}_\xi(\pi_d X_1) \underline{\sim} \sigma^2_d(\xi)/N$, in certain senses as $N \rightarrow \infty$, where $\xi_{0, d}$ is an asymptotically stable fixed point of $\pi_d\mathbf{f}_N(\xi) = \xi$ and $x_0 = \lim_{d \rightarrow \infty} \xi_{0, d}$ exists in $\mathbf{P}\lbrack 0, 1\rbrack$. Assuming various regularity conditions and $\beta = \beta_N \rightarrow 0, N\beta/\log N \rightarrow \infty$, it is shown that the expected time it takes the Markov chain to exit a fixed open ball $D$ about $x_0$ once $X_0 \in D$ is logarithmically equivalent to $\exp\lbrack N\beta_N V \rbrack$, where $V > 0$ is a limit of solutions $V_d = V_d(h_d, \sigma_d)$ of variational problems of Wentzell-Freidlin type in $\mathbb{R}^d$ as $d \rightarrow \infty$. These results apply to an infinite alleles model in population genetics, where $\{X_n\}$ represents the evolution of distributions of types among a population of $N$ randomly mating genes, and where forces of mutation and selection are stronger than effects due to finite population size.