First Passage of a Randomly Accelerated Particle

TL;DR

This paper reviews the first-passage and first-exit properties of a randomly accelerated particle, exploring various boundary conditions, return statistics, and applications, highlighting the process's complex stochastic behavior.

Contribution

It provides a comprehensive review of first-passage and first-exit statistics for the random acceleration process, including new insights into boundary conditions and applications.

Findings

Analysis of first arrival and return times at the origin.

First-exit properties from finite intervals.

Connections between extreme value and first-passage statistics.

Abstract

In the random acceleration process, a point particle is accelerated according to $\ddot{x}=\eta(t)$, where the right hand side represents Gaussian white noise with zero mean. We begin with the case of a particle with initial position $x_0$ and initial velocity $v_0$ and review the statistics of its first arrival at the origin and its first return to the origin. Multiple returns to the origin, motion with a constant force in addition to a random force, and persistence properties for several boundary conditions at the origin are also considered. Next we review first-exit properties of a randomly accelerated particle from the finite interval $0<x<1$. Then the close connection between the extreme value statistics of a randomly accelerated particle and its first-passage properties is discussed. Finally some applications where first-passage statistics of the random acceleration process play a role are considered.