Let $(X_{n})_{n\in\mathbb{N}_{0}}$ be an irreducible Markov chain, i.e. every state in the state space leads to every state in the state space. Moreover, let the state space be the non-negative integers.
Is there a way to set up a of system of equations, or maybe a sequence of system of equations to determine whether or not the state $0$ is recurrent?
If so, how would such a thing look like?
Here we say that $x$ leads to $y$ iff $\rho_{xy}:=\Pr_{x}(T_{y}<\infty)>0$, where $T_{y}=\inf\{n, n>0\text{ such that } X_{n}=y\}$. We also say that $0$ is recurrent iff $\rho_{00}:=\Pr_{0}(T_{0}<\infty)=1$.
My attempt so far:
Since the chain is irreducible, either all states are recurrent or all states are transient, i.e. $\rho_{xy}=1$ or $\rho_{xy}<1$ for all $x,y\in\mathbb{N}$ respectively...