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.

v3=final version (43 pages) ; v2=revised version (43 pages) with a lot of new material with respect to the initial version v1 (19 pages)

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• Stochastic resetting in discrete-time quantum dynamics: steady states and correlations in few-qubit systems
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• Large deviations for trajectory observables of diffusion processes in dimension $d>1$ in the double limit of large time and small diffusion coefficient
• A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains
• Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?
• Thermodynamics of quantum-jump trajectories of open quantum systems subject to stochastic resetting
• Conditioning two diffusion processes with respect to their first-encounter properties
• Classification of diffusion processes in dimension $d$ via the Carleman approach with applications to models involving additive, multiplicative or square-root noises