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Optimal non-Gaussian search with stochastic resetting.
Aleksander Stanislavsky1,2, Aleksander Weron2
1Institute of Radio Astronomy, 4 Mystetstv Street, 61002 Kharkiv, Ukraine.
Physical Review. E
|August 20, 2021
Summary
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.
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.
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