Time between the maximum and the minimum of a stochastic process

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 Lévy flights and find that it differs from the Brownian motion result.

Main text (published version): 5 pages + 3 Figs, Supp. Mat.: 20 pages + 7 Figs, typos corrected