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.

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

“…In these studies, as explained in detail in the two complementary papers [50,51] and in the HDR thesis [52], the Doob conditioned processes correspond to the processes that optimize the dynamical large deviations in the presence of the imposed constraints, showing the link with the field of stochastic control. It should be stressed that the corresponding rate functions at Level 2.5 are explicit for many Markov processes, including discrete-time Markov chains [72][73][74][75][76], continuous-time Markov jump processes [52,72, and Diffusion processes [52,65,75,76,80,81,93,96,97]. As incredible as it may seem, the very deep connections between the fields of Doob conditioning, of large deviations and of stochastic control are actually already present in the famous paper written in 1931 by E. Schrödinger [98], as discussed in detail in the recent detailed commentary [99] accompanying its english translation, as well as in the two reviews [100,101] written from the viewpoint of stochastic control and optimal transport.…”

“…Within the standard classification of the large deviation theory (see the reviews [115][116][117] and references therein), the Level 2 concerning the empirical density alone is usually not closed for non-equilibrium processes with steady currents, while the Level 3 concerning the whole empirical process is somewhat too general. 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 [72][73][74][75][76]117], continuous-time Markov jump processes [52,72, and Diffusion processes [52,65,75,76,80,81,93,96,97]. This explains why the Level 2.5 can be considered as the cornerstone from which many other dynamical large deviations properties can be derived via the appropriate contractions.…”

Conditioning two diffusion processes with respect to their first-encounter properties

“…Within the classification into levels of the large deviation theory (see the reviews [94][95][96] and references therein), the Level 2 concerning the empirical density alone is usually not closed for non-equilibrium processes that do not satisfy detailed balance. However, the Level 2.5 concerning 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 [96,[106][107][108][109][110], continuous-time Markov jump processes [57,106, and diffusion processes [57,70,109,110,114,115,127,130,131]. This explains why the Level 2.5 has been a major progress in the field of dynamical large deviations properties for Markov processes.…”

“…(., ., .) at Level 2.5 is given by the usual explicit form for diffusion processes [57,70,109,110,114,115,127,130,131] applied to our present case I…”

Effect of perturbations on systems near extinction

“…Furthermore, the second eigenvalue λ 1 has to be real and the corresponding left and right eigenvectors are real with positive components in the interior [See Supplementary Information for more details [36]. See also [37]]. The orthogonality between left and right eigenvectors leads to…”