# Archive ouverte HAL – Time Between the Maximum and the Minimum of a Stochastic Process

### Francesco Mori 1 Satya N. Majumdar 1 Satya Majumdar 1 Gregory Schehr 1

#### Francesco Mori, Satya N. Majumdar, Satya Majumdar, Gregory Schehr. Time Between the Maximum and the Minimum of a Stochastic Process. Physical Review Letters, American Physical Society, 2019, 123 (20), ⟨10.1103/PhysRevLett.123.200201⟩. ⟨hal-02395492⟩

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.

• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques