First-Passage and Extreme-Value Statistics of a Particle Subject to a Constant Force Plus a Random Force

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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 x1 of a particle which starts at x0 with velocity v0. The probability that the particle has not yet arrived at x1 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)].

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