Time between the maximum and the minimum of a stochastic process

TL;DR

This paper derives the exact probability density function for the time difference between the maximum and minimum of a Brownian motion and its generalizations, with applications to interface height fluctuations and universality considerations.

Contribution

It provides an exact solution for the distribution of the time difference between extrema of Brownian motion and extends it to Brownian bridges and other processes.

Findings

Exact distribution for Brownian motion and bridge cases.

Universal asymptotic behavior for finite-variance random walks.

Numerical results showing differences for Lévy flights.

Abstract

We present an exact solution for the probability density function $P(\tau=t_{\min}-t_{\max}|T)$ of the time-difference between the minimum and the maximum of a one-dimensional Brownian motion of duration $T$. We then generalise 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 height 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 L\'evy flights and find that it differs from the Brownian motion result.