The following is a fact about Markov chains that came up in a game theory paper. The purpose of this question is to ask if related notions or similar results are found elsewhere in probability, or are of interest to probabilists.
Fix a finite state space $S$. Let $K(\cdot,\cdot)$ denote the transition kernel of an ergodic, reversible Markov process on $S$, and $\pi$ the associated ergodic distribution. A set $E \subseteq S$ is called $p$-cohesive if, for every $x \in E$, we have $K(x,E)\geq p$: the probability of staying within $E$ conditional on being in it is at least $p$.
Proposition. Fix $p<1/2$ and $\delta >0$. There is an $\epsilon>0$ such that, for any kernel $K$ and set $E$ so that $\pi(E)\geq 1-\epsilon$, there is a $p$-cohesive $F \subseteq E$ with $\pi(F)\geq 1-\delta$. Thus, if in the long run the chain spends a lot of time in $E$, then there is some subset of $E$ in which it also spends a lot of time, and from which the exit probability is bounded.
This is tight in the sense that for any $p> 1/2$, there is an ergodic, reversible Markov chain with no proper, nonempty $p$-cohesive sets (e.g. symmetric random walk on the integers, reflected at two boundaries).
I would be very grateful for pointers on things to read to find connections in the probability literature.