Distribution of the time of the maximum for stationary processes

Abstract

We consider a one-dimensional stationary stochastic process $x(\tau )$ of duration $T$. We study the probability density function (PDF) $P(t_{\mathrm{m}}|T)$ of the time $t_{\mathrm{m}}$ at which $x(\tau )$ reaches its global maximum. By using a path integral method, we compute $P(t_{\mathrm{m}}|T)$ for a number of equilibrium and nonequilibrium stationary processes, including the Ornstein-Uhlenbeck process, Brownian motion with stochastic resetting and a single confined run-and-tumble particle. For a large class of equilibrium stationary processes that correspond to diffusion in a confining potential, we show that the scaled distribution $P(t_{\mathrm{m}}|T)$, for large $T$, has a universal form (independent of the details of the potential). This universal distribution is uniform in the ``bulk'', i.e., for $0\ll t_{\mathrm{m}}\ll T$ and has a nontrivial edge scaling behavior for $t_{\mathrm{m}}\to 0$ (and when $t_{\mathrm{m}}\to T$), that we compute exactly. Moreover, we show that for any equilibrium process the PDF $P(t_{\mathrm{m}}|T)$ is symmetric around $t_{\mathrm{m}}=T/2$, i.e., $P(t_{\mathrm{m}}|T)=P(T-t_{\mathrm{m}}|T)$. This symmetry provides a simple method to decide whether a given stationary time series $x(\tau )$ is at equilibrium or not.

Main text: 6 pages + 4 figs., Supp. Mat.: 7 pages + 3 figs

References in corpus (13)

• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Diffusive transport without detailed balance in motile bacteria: Does microbiology need statistical physics?
• Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results
• Probability currents as principal characteristics in the statistical mechanics of non-equilibrium steady states
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Spatial extent of an outbreak in animal epidemics
• On the time to reach maximum for a variety of constrained Brownian motions
• Safe Leads and Lead Changes in Competitive Team Sports
• Generalized arcsine laws for fractional Brownian motion
• Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
• A Heterogeneous Out-of-Equilibrium Nonlinear $q$-Voter Model with Zealotry
• Nonequilibrium oscillations, probability angular momentum, and the climate system
• Exact results for a simple epidemic model on a directed network: Explorations of a system in a non-equilibrium steady state

Cited by in corpus (5)

• Number of distinct sites visited by a resetting random walker
• Inference in non-equilibrium systems from incomplete information: the case of linear systems and its pitfalls
• Extreme value statistics and arcsine laws for heterogeneous diffusion processes
• Discrete Sampling of Extreme Events Modifies Their Statistics
• Inferring entropy production rate from partially observed Langevin dynamics under coarse-graining