Large deviations for Markov processes with stochastic resetting: analysis via the empirical density and flows or via excursions between resets

Abstract

Markov processes with stochastic resetting towards the origin generically converge towards non-equilibrium steady-states. Long dynamical trajectories can be thus analyzed via the large deviations at level 2.5 for the joint probability of the empirical density and the empirical flows, or via the large deviations of semi-Markov processes for the empirical density of excursions between consecutive resets. The large deviations properties of general time-additive observables involving the position and the increments of the dynamical trajectory are then analyzed in terms of the appropriate Markov tilted processes and of the corresponding conditioned processes obtained via the generalization of Doob’s h-transform. This general formalism is described in detail for the three possible frameworks, namely discrete-time/discrete-space Markov chains, continuous-time/discrete-space Markov jump processes and continuous-time/continuous-space diffusion processes, and is illustrated with explicit results for the Sisyphus random walk and its variants, when the reset probabilities or reset rates are space-dependent.

Thermodynamics of quantum-jump trajectories of open quantum systems subject to stochastic resetting

“…[50]. The renewal approach is effective also for the computation of the large deviation statistics of dynamical, i.e., time-integrated, observables of the resetting dynamics [51][52][53][54]. In Refs.…”

Microcanonical conditioning of Markov processes on time-additive observables

“…The 'canonical conditioning' (see the reminder in the two Appendices) of the Sisyphus Markov jump process has been studied in [7] for the more general case where the reset rates of the initial model are space-dependent r x (instead of being given by the constant value r) and where the time-additive observable involve two arbitrary functions α(x) and β(x, x ).…”

“…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.…”

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.…”