I was wondering about the following problem:

Assume we have a state space $S:=\mathbb{Z}$ and a Markov chain, such that we can go from any state $x$ to some state $y$ with positive probabilities, i.e. $p_t(x,y)>0$ for any $t >0 $ and $x,y \in S.$

Let $T_0^x$ be the hitting time to go from state $x$ to $0$.

If we know that $P(\lim_{x \rightarrow -\infty} T_0^x<\infty)=1.$ So we can almost surely go to state $0$ from -infinity. Does this imply that $\liminf_{x \rightarrow -\infty} p_t(x,0)>0,$ for all $t>0$ i.e. does this imply that we go to state $0$ in every finite time step?

The thing is that we a priori only know that we can go to $0$ in some finite time, but not necessarily in any finite time (with some positive probability.)