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The large deviations at Level 2.5 are applied to Markov processes withabsorbing states in order to obtain the explicit extinction rate of metastablequasi-stationary states in terms of their empirical time-averaged density andof their time-averaged empirical flows over a large time-window $T$. Thestandard spectral problem for the slowest relaxation mode can be recovered fromthe full optimization of the extinction rate over all these empiricalobservables and the equivalence can be understood via the Doob generator of theprocess conditioned to survive up to time $T$. The large deviation propertiesof any time-additive observable of the Markov trajectory before extinction canbe derived from the Level 2.5 via the decomposition of the time-additiveobservable in terms of the empirical density and the empirical flows. Thisgeneral formalism is described for continuous-time Markov chains, withapplications to population birth-death model in a stable or in a switchingenvironment, and for diffusion processes in dimension $d$.

journal of statistical mechanics

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