First-passage and extreme-value statistics of a particle subject to a constant force plus a random force

TL;DR

This paper investigates the first-passage, return, and extreme-value statistics of a particle under a constant and random Gaussian white noise force, providing analytical results for various related probabilities and distributions.

Contribution

It derives analytical expressions for first-passage times, return times, and maximum displacement distributions for a particle influenced by combined constant and stochastic forces.

Findings

Derived the probability distribution of first arrival times.

Calculated the mean first arrival time and velocity distribution at arrival.

Established the relationship between extreme-value and first-passage statistics.

Abstract

We consider a particle which moves on the x axis and is subject to a constant force, such as gravity, plus a random force in the form of Gaussian white noise. We analyze the statistics of first arrival at point $x_1$ of a particle which starts at $x_0$ with velocity $v_0$. The probability that the particle has not yet arrived at $x_1$ after a time $t$, the mean time of first arrival, and the velocity distribution at first arrival are all considered. We also study the statistics of the first return of the particle to its starting point. Finally, we point out that the extreme-value statistics of the particle and the first-passage statistics are closely related, and we derive the distribution of the maximum displacement $m={\rm max}_t[x(t)]$.