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

TL;DR

This paper develops a comprehensive large deviation framework for Markov processes with stochastic resetting, analyzing non-equilibrium steady states and trajectory fluctuations across different process types.

Contribution

It introduces a unified formalism for large deviations in Markov processes with resets, covering discrete and continuous spaces and times, with explicit examples.

Findings

Explicit large deviation rate functions for reset processes

Analysis of time-additive observables via tilted and conditioned processes

Application to Sisyphus Random Walk variants with space-dependent 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.