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Stochastic representation of processes with resetting.

Marcin Magdziarz1, Kacper Taźbierski1

  • 1Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland.

Physical Review. E
|August 17, 2022
PubMed
Summary
This summary is machine-generated.

We present a new method to model stochastic processes that restart, using jump-diffusion models. This framework analyzes Brownian motion with resetting, offering a general approach for complex systems.

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Area of Science:

  • Stochastic processes
  • Mathematical physics
  • Statistical mechanics

Background:

  • Stochastic processes are fundamental in modeling complex systems.
  • Resetting mechanisms are crucial in various physical and biological phenomena.
  • Existing models often lack a general framework for analyzing resetting processes.

Purpose of the Study:

  • To introduce a general stochastic representation for processes with resetting.
  • To provide a unified framework for analyzing stochastic processes with intermittent restarts.
  • To extend the analysis to both random and non-random resetting points.

Main Methods:

  • Utilizing stochastic differential equations, specifically jump-diffusion models.
  • Developing analytical techniques for processes with resetting.
  • Employing Monte Carlo simulation methods for analysis.

Main Results:

  • Derivation of fundamental properties for Brownian motion with Poissonian resetting (e.g., Itô lemma, Fokker-Planck equations).
  • Explicit forms for probability density functions and moments of all orders.
  • Extension of results to time-nonhomogeneous Poissonian resetting.

Conclusions:

  • The proposed jump-diffusion framework offers a versatile tool for analyzing stochastic processes with resetting.
  • This approach enables both analytical and simulation-based studies of complex systems.
  • The framework provides a comprehensive understanding of systems with intermittent random resetting.