Definition (Recurrent)

A set $A\subset \mathbb{X}$ is said to be recurrent if the Markov chain visits $A$ infinitely often in expectation, when the process starts in $A$: $$\mathbb{E}_{X}[\eta_{A}]=\mathbb{E}[\eta_{A}|X_{0}=X]=\infty, \ \ \forall x\in A$$ where $\eta_{A}$ is the Occupation Time.