Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment

The large deviations at Level 2.5 are applied to Markov processes with absorbing states in order obtain the explicit extinction rate of metastable quasi-stationary terms their empirical time-averaged density and flows over a time-window $T$. standard spectral problem for slowest relaxation mode can be recovered from full optimization all these observables equivalence understood via Doob generator process conditioned survive up time deviation properties any time-additive observable trajectory before derived decomposition flows. This general formalism is described continuous-time chains, applications population birth-death model stable or switching environment, diffusion dimension $d$.