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- W3185473721 abstract "We consider a one-dimensional stationary stochastic process $x(tau)$ of duration $T$. We study the probability density function (PDF) $P(t_{rm m}|T)$ of the time $t_{rm m}$ at which $x(tau)$ reaches its global maximum. By using a path integral method, we compute $P(t_{rm m}|T)$ 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 $P(t_{rm m}|T)$, 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 $0 ll t_{rm m} ll T$ and has a nontrivial edge scaling behavior for $t_{rm m} to 0$ (and when $t_{rm m} to T$), that we compute exactly. Moreover, we show that for any equilibrium process the PDF $P(t_{rm m}|T)$ is symmetric around $t_{rm m}=T/2$, i.e., $P(t_{rm m}|T)=P(T-t_{rm m}|T)$. This symmetry provides a simple method to decide whether a given stationary time series $x(tau)$ is at equilibrium or not." @default.
- W3185473721 created "2021-08-02" @default.
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- W3185473721 date "2021-04-15" @default.
- W3185473721 modified "2023-09-25" @default.
- W3185473721 title "Distribution of the time of the maximum for stationary processes" @default.
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