Time Between the Maximum and the Minimum of a Stochastic Process

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 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.

Figure 1

Typical trajectory of Brownian motion $x(t)$ during time interval [0, $T$], starting from $x(0)=0$. The global maximum $x_{\max }$ occurs at time $t_{\max }$ and the global minimum $-x_{\min }$ at $t_{\min }$. The final position $x(T)$, measured with respect to $-x_{\min }\le 0$, is denoted by $x_{\mathrm{f}}$. The total time interval [0, $T$] is divided into three segments of $[0,t_{\max }]$ (I), $[t_{\max },t_{\min }]$ (II), and $[t_{\min },T]$ (III) for the case of $t_{\min }>t_{\max }$.

Figure 2

Scaled distribution $TP(\tau |T)$ plotted as a function of $\tau /T$ for BM (solid line corresponds to exact scaling function $f_{\mathrm{BM}}(y)$ in Eq. (2), whereas filled dots are the results of simulations). Inset: the same scaled distribution for the Brownian bridge where the exact scaling function $f_{\mathrm{BB}}(y)$ is given in Eq. (4).

Figure 3

Scaled distribution $nP(\tau |n)$ vs $\tau /n$ for random walks (RWs) with different jump distributions. The Gaussian, uniform, double-exponential, and Pareto (with index $\mu =3$) jump distributions, all of which have a finite variance, collapse onto the Brownian scaling function $f_{\mathrm{BM}}(y)$ shown by the solid (green) line. For Lévy flights with index $\mu =3/2$ and $\mu =1$ (Cauchy distribution), the scaling function $f_{\mu }(y)$ depends on $\mu$ (except at the endpoints $y=\pm 1$, where $f_{\mu }(\pm 1)=1/2$ seems to be universal for all $0<\mu \le 2$).