We construct Cheeger-type bounds for the second eigenvalue of a substochastic transition probability matrix in terms of the Markov chain's conductance and metastability (and vice versa) with respect to its quasistationary distribution, extending classical results for stochastic transition matrices.
"Cheeger inequalities for absorbing Markov chains." Adv. in Appl. Probab. 48 (3) 631 - 647, September 2016.