Time Between the Maximum and the Minimum of a Stochastic Process

Francesco Mori, Satya N. Majumdar, and Grégory Schehr
Phys. Rev. Lett. 123, 200201 – Published 15 November 2019
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Abstract

We present an exact solution for the probability density function P(τ=tmintmax|T) of the time difference between the minimum and the maximum of a one-dimensional Brownian motion of duration T. We then generalize our results to a Brownian bridge, i.e., a periodic Brownian motion of period T. We demonstrate that these results can be directly applied to study the position difference between the minimal and the maximal heights of a fluctuating (1+1)-dimensional Kardar-Parisi-Zhang interface on a substrate of size L, in its stationary state. We show that the Brownian motion result is universal and, asymptotically, holds for any discrete-time random walk with a finite jump variance. We also compute this distribution numerically for Lévy flights and find that it differs from the Brownian motion result.

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  • Received 13 September 2019

DOI:https://doi.org/10.1103/PhysRevLett.123.200201

© 2019 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

Francesco Mori, Satya N. Majumdar, and Grégory Schehr

  • LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France

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Issue

Vol. 123, Iss. 20 — 15 November 2019

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