Let $(X_n)_{n \geq 0}$ be a irreducible Markov chain defined on a countable state space $S$. It is known some ways to figure out if this chain is recurrent or not looking for superharmonic functions with special properties.
For example, using martingale theory, if there is a non-constant, non-negative function defined on $S$ such that it is a superharmonic function with respect the transition probabilities of this chain then we can conclude transience. Another example, if there is a finite set $F \subset S$ and a superharmonic function on $S\setminus F$, $\phi : S \longrightarrow [0,+\infty)$, such that $\{\phi \leq M\}$ is a finite set for every $M>0$ then the chain is recurrent.
My question: is there a way to know if the chain is null recurrent using these kind of techniques?
Thanks in advance!