If $T$ is a row stochastic matrix (finite or infinite) and $x,y$ are row vectors in $\mathbb{R}^n_+$ (or $\ell^1_+$) then $d_H(xT,yT) \leq d_H(x,y)$ where
$$d_H(x,y)=\log \frac{\sup_i x_i/y_i}{\inf_i x_i/y_i}.$$
This says that the operator $x \mapsto xT$ is non-expansive in the Hilbert metric $d_H$. In finite dimensions that immediately implies that a stationary distribution exists, and one can use various improvements of this observation under additional assumptions to extract things like estimates of ergodicity coefficients and so on. This might be called "Birkhoff's Perron-Frobenius program".
In infinite dimensions things are necessarily more complicated because transient Markov chains exist on such spaces. I'm trying to see if there is any good way to use this observation or improvements of it to study the stability of Markov chains on countable state spaces. It seems the obvious source to look at is Non-negative Matrices and Markov chains by Eugene Seneta, but in the section on countable matrices he does not go in this direction. Do you know of any developments in this direction?
For some context, I have a parametrized family of Markov chains on a countable state space. For small enough values of one of the parameters, the chain is positive recurrent; for large enough values it is transient. I'm just trying to find or estimate the boundary between these regimes, and I'm finding this quite difficult. So I'm trying to see if there is anything I can do besides the usual Lyapunov-Foster approach.