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

Abstract

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

Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically-constrained East Model

“…While the initial classification involved only three nested levels (see the reviews [17][18][19] and references therein), with Level 1 for empirical observables, Level 2 for the empirical density, and Level 3 for the empirical process, the introduction of the Level 2.5 has been a major progress in order to characterize the joint distribution of the empirical density and of the empirical flows. Its essential advantage is that the rate functions at Level 2.5 can be written explicitly for general Markov processes, including discrete-time Markov chains [19][20][21][22][23][24], continuous-time Markov jump processes [20,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] and Diffusion processes [23,24,28,29,32,[42][43][44][45]. As a consequence, the explicit Level 2.5 can be considered as the central starting point from which many other large deviations properties can be derived via the appropriate contraction.…”

Microcanonical conditioning of Markov processes on time-additive observables

“…For instance, the large deviations at Level 2 for the empirical density allows to analyze the time-additive observables that only depend on the time spent in each configuration, but the Level 2 is usually not closed for non-equilibrium processes with steady currents. By contrast, the Level 2.5 concerning the joint distribution of the empirical density and of the empirical flows can be written in closed form for general Markov processes, including discrete-time Markov chains [3][4][5][6][7][8], continuoustime Markov jump processes [4, and Diffusion processes [7,8,12,13,16,26,[29][30][31]. In addition, this Level 2.5 is necessary to analyze via contraction the general case of time-additive observables that involve not only the time spent in each configuration but also the elementary increments of the Markov process.…”

Inhomogeneous asymmetric exclusion processes between two reservoirs: large deviations for the local empirical observables in the mean-field approximation

“…(a) On one hand, one can consider the Markov process in the space of configurations (for instance 2 N configurations for a system of N classical spins) and one can apply the explicit large deviations at level 2.5 to characterize the joint distribution of the empirical density and of the empirical flows in the configuration space. Indeed, while the initial classification involved only three levels (see the reviews [1][2][3] and references therein), with level 1 for empirical observables, level 2 for the empirical density, and level 3 for the empirical process, the introduction of the level 2.5 has been a major progress to characterize non-equilibrium steady states, because the rate functions at level 2.5 can be written explicitly for general Markov processes, including discrete-time Markov chains [3][4][5][6][7][8], continuous-time Markov jump processes [4, and diffusion processes [7,8,12,13,16,25,26,[28][29][30][31]. From this explicit level 2.5, many other large deviations properties can be derived via the appropriate contraction.…”