Gaussian Process and Levy Walk under Stochastic Non-instantaneous Resetting and Stochastic Rest

TL;DR

This paper models a stochastic process combining Gaussian and Levy walk dynamics with non-instantaneous resetting and rest phases, analyzing their effects on displacement, stationary states, and first passage times.

Contribution

It introduces a comprehensive model integrating Gaussian and Levy walks with stochastic resetting and resting, providing new insights into their combined asymptotic behaviors.

Findings

Derived asymptotic behaviors of mean squared displacement.

Analyzed stationary distributions for localized processes.

Calculated mean first passage times for Brownian motion dynamics.

Abstract

A stochastic process with movement, return, and rest phases is considered in this paper. For the movement phase, the particles move following the dynamics of Gaussian process or ballistic type of L\'evy walk, and the time of each movement is random. For the return phase, the particles will move back to the origin with a constant velocity or acceleration or under the action of a harmonic force after each movement, so that this phase can also be treated as a non-instantaneous resetting. After each return, a rest with a random time at the origin follows. The asymptotic behaviors of the mean squared displacements with different kinds of movement dynamics, random resting time, and returning are discussed. The stationary distributions are also considered when the process is localized. Besides, the mean first passage time is considered when the dynamic of movement phase is Brownian motion.