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Poissonian resetting of subordinated Brownian motion leads to a stationary Laplace distribution. Optimal resetting rates minimize the mean time to reach a target, a key finding for stochastic processes.

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Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Mathematical Physics

Background:

  • Subordinated Brownian motion exhibits subdiffusion.
  • Stochastic processes with resetting are crucial for modeling various physical phenomena.
  • Understanding stationary states is fundamental in statistical mechanics.

Purpose of the Study:

  • To determine the stationary state distribution of subordinated Brownian motion under Poissonian resetting.
  • To analyze the impact of resetting on the mean time to reach a target.
  • To investigate the stationary state for Lévy motion (superdiffusion) with resetting.

Main Methods:

  • Analysis of subordinated Brownian processes with Poissonian resetting.
  • Derivation and characterization of stationary state distributions (Laplace and Linnik).
  • Investigation of the mean first passage time and its dependence on resetting rate.

Main Results:

  • The stationary state for subdiffusion under resetting is a Laplace distribution.
  • The location parameter depends on the reset position; the scaling parameter depends on the parent process's Laplace exponent.
  • A finite, minimum mean time to reach a target exists, dependent on the resetting rate.
  • Superdiffusion (Lévy motion) under resetting leads to a Linnik distribution.

Conclusions:

  • Poissonian resetting leads to predictable stationary states (Laplace for subdiffusion, Linnik for superdiffusion).
  • The analysis allows restoration of probability density functions from scaling parameters.
  • An optimal resetting rate exists that minimizes the mean arrival time, relevant for transport phenomena.