Time to reach the maximum for a stationary stochastic process

Abstract

We consider a one-dimensional stationary time series of fixed duration $T$. We investigate the time $t_{\mathrm{m}}$ at which the process reaches the global maximum within the time interval $[0,T]$. By using a path-decomposition technique, we compute the probability density function $P(t_{\mathrm{m}}|T)$ of $t_{\mathrm{m}}$ for several processes, that are either at equilibrium (such as the Ornstein-Uhlenbeck process) or out of equilibrium (such as Brownian motion with stochastic resetting). We show that for equilibrium processes the distribution of $P(t_{\mathrm{m}}|T)$ is always symmetric around the midpoint $t_{\mathrm{m}}=T/2$, as a consequence of the time-reversal symmetry. This property can be used to detect nonequilibrium fluctuations in stationary time series. Moreover, for a diffusive particle in a confining potential, we show that the scaled distribution $P(t_{\mathrm{m}}|T)$ becomes universal, i.e., independent of the details of the potential, at late times. This distribution $P(t_{\mathrm{m}}|T)$ becomes uniform in the "bulk" $1\ll t_{\mathrm{m}}\ll T$ and has a nontrivial universal shape in the "edge regimes" $t_{\mathrm{m}}\to 0$ and $t_{\mathrm{m}}\to T$. Some of these results have been announced in a recent Letter [Europhys. Lett. {\bf 135}, 30003 (2021)].

arXiv comment57 pages, 19 figures. This is a longer version of arXiv:2104.07346, published in Europhysics Letters

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