I'm reading this notes from Martin Hairer about convergence of Markov Processes (on a discrete state space $S$ and in continuous time). On page 12, before presenting the so-called "Harris Theorem", the following assumption is made, toghether with an heuristic explanation:

For a fixed $T>0$, there exists a function $V:S\to [0,\infty)$ and constants $K\geqslant 0$ and $\gamma\in (0,1)$ such that $$ \mathcal{P}_TV(x)\leqslant \gamma V(x) +K $$ Where $ \mathcal{P}_Tf(x)=\mathbb{E}_x(f(X_T))$ , i.e., $ \mathcal{P}$ is the Markov semigroup of the MC $(X_t)$ .

The heuristic idea of the above assumption is this one:

[The above assumption] ensures that the dynamic enters the “centre” of the state space regularly with tight control on the length of the excursions from the centre.

So my questions are:

a) Can this idea be mathematically presented?

b) Suppose $S$ is $\mathbb{N}_0^n$, with $n\in \mathbb{N}$ , and let $\mathcal{C}$ be some finite neighbourhood of the origin. Then for $x\notin \mathcal{C}$ and $T_{\mathcal{C}}$ the hitting time of $\mathcal{C}$, can I assure that $$ \mathbb{P}_x(T_{\mathcal{C}}<\infty)=1 $$

c) What about question **b)** if I know that $V(x)=||x||_1:=\sum|x_i|$ and that $K=0$?

This question arises in the context of a problem whith this Markov Chain.

As for question **c)**, I've only been able to say that

$$\mathbb{P}_x(||X_T||<||x||_1)\leqslant \gamma $$

with $T$ and $\gamma$ as before.