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Large deviations for Markov processes with resetting.

Janusz M Meylahn1,2, Sanjib Sabhapandit3, Hugo Touchette2,4

  • 1Mathematical Institute, Leiden University, Leiden, The Netherlands.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2016
PubMed
Summary
This summary is machine-generated.

We analyze large deviations for Markov processes with random resetting. A renewal formula helps determine the rate function for time-additive observables, crucial for understanding random search and population dynamics.

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Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Probability Theory

Background:

  • Markov processes with random resetting are increasingly studied for applications in random searches, foraging, and population dynamics.
  • Understanding the fluctuations of time-additive observables in these systems is essential for modeling their long-term behavior.

Purpose of the Study:

  • To investigate the large deviations of time-additive functions for Markov processes that incorporate random resetting.
  • To derive a general method for calculating the rate function that characterizes the fluctuations of these observables.

Main Methods:

  • A renewal formula was derived to connect generating functions of processes with and without resetting.
  • This formula was used to obtain the rate function for time-additive observables in resetting Markov processes.
  • The large deviations of the area under the Ornstein-Uhlenbeck process with resetting were analyzed as a specific example.

Main Results:

  • The study provides a method to calculate the rate function, which quantifies the probability of observing large deviations in observables.
  • The derived renewal formula offers a unified approach to analyze resetting effects on stochastic processes.
  • The analysis of the Ornstein-Uhlenbeck process demonstrates the practical application of the method.

Conclusions:

  • The developed framework allows for the characterization of large deviations in a broad class of resetting Markov processes.
  • This work contributes to the theoretical understanding of stochastic processes with resetting, with implications for various scientific fields.
  • The findings are applicable to diverse systems including diffusions, random walks, and jump processes with resetting mechanisms.