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

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 $m={\max }_t[x(t)]$.

Contains an analysis of the extreme-value statistics not included in first version

References in corpus (4)

• Free energy and extension of a semiflexible polymer in cylindrical confining geometries
• Maximal height statistics for 1/f^alpha signals
• Extreme statistics for time series: Distribution of the maximum relative to the initial value
• Equilibrium statistics of an inelastically bouncing ball, subject to gravity and a random force

Cited by in corpus (6)

• Time to reach the maximum for a random acceleration process
• Random acceleration process under stochastic resetting
• Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids
• The convex hull for a random acceleration process in two dimensions
• Statistics of the first passage time of Brownian motion conditioned by maximum value or area
• Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier