Large deviations for Markov processes with resetting

TL;DR

This paper investigates the probabilities of rare fluctuations in Markov processes that are periodically reset, providing a mathematical framework to understand their long-term behavior and applications in various stochastic models.

Contribution

The authors derive a renewal formula linking generating functions with and without resetting, enabling the calculation of large deviation rate functions for observables of reset Markov processes.

Findings

Derived a renewal formula for generating functions with resetting.

Calculated the large deviation rate function for the area of an Ornstein-Uhlenbeck process with resetting.

Discussed applications to diffusions, random walks, and jump processes with resetting.

Abstract

Markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches, foraging, and population dynamics. Here we study the large deviations of time-additive functions or observables of Markov processes with resetting. By deriving a renewal formula linking generating functions with and without resetting we are able to obtain the rate function of such observables, characterizing the likelihood of their fluctuations in the long-time limit. We consider as an illustration the large deviations of the area of the Ornstein-Uhlenbeck process with resetting. Other applications involving diffusions, random walks, and jump processes with resetting or catastrophes are discussed.