Classical arcsine law states that fraction of occupation time on the positive or the negative side in Brownian motion does not converge to a constant but converges in distribution to the arcsine distribution. Here, we consider how a preparation of the system affects the arcsine law, i.e., aging of the arcsine law. We derive aging distributional theorem for occupation time statistics in Brownian motion, where the ratio of time when measurements start to the measurement time plays an important role in determining the shape of the distribution. Furthermore, we show that this result can be generalized as aging distributional limit theorem in renewal processes.
We give a general proof of aging for trap models using the arcsine law for stable subordinators. This proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape. We apply this proof to aging for trap models on large two-dimensional tori and for trap dynamics of the Random Energy Model on a broad range of time scales.
Applying the new tools developed in [G1], we investigate the arcsine aging regime of the random hopping time dynamics of the REM. Our results are optimal in several ways. They cover the full time-scale and temperature domain where this phenomenon occurs. On this domain the limiting clock process and associated time correlation function are explicitly constructed. Finally, all convergence statements w.r.t. the law of the random environment are obtained in the strongest sense possible, except perhaps on the very last scales before equilibrium.
In this paper the celebrated arcsine aging scheme of G. Ben Arous and J. Cerný is taken up. Using a brand new approach based on point processes and weak convergence techniques, this scheme is implemented in a wide class of Markov processes that can best be described as Glauber dynamics of discrete disordered systems. More specifically, conditions are given for the underlying clock process (a partial sum process that measures the total time elapsed along paths of a given length) to converge to a subordinator, and this subordinator is constructed explicitly. This approach is illustrated on Bouch…
The gaps between the times, in a Weyl chamber, at which the sum of the increments of independent Brownian motions attains its maximum has a Dirichlet distribution.
We identify the limit of the internal DLA cluster generated by Sinai's walk as the law of a functional of a Brownian motion which turns out to be a new interpretation of the Arcsine law.
We prove a geometric extension of Lévy's occupation time arcsine law near a hypersurface on a Riemannian manifold. The deviation from the classic arcsine law is of the order of the square-root of the time horizon and is expressed explicitly in terms of the mean curvature of the hypersurface and the local time of the underlying standard Brownian motion.
We consider one-dimensional directed trap models and suppose that the trapping times are heavy-tailed. We obtain the inverse of a stable subordinator as scaling limit and prove an aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality described by Ben Arous and Černý for a large class of graphs.
The additive monotone (resp. boolean) unitary Brownian motion is a non-commutative stochastic process with monotone (resp. boolean) independent and stationary increments which are distributed according to the arcsine law (resp. Bernoulli law) . We introduce the monotone and booleen unitary Brownian motions and we derive a closed formula for their associated moments. This provides a description of their spectral measures. We prove that, in the monotone case, the multiplicative analog of the arcsine distribution is absolutely continuous with respect to the Haar measure on the unit circle, wherea…
We show that the last zero before time $t$ of a recurrent Bessel process with drift starting at $0$ has the same distribution as the product of an independent right censored exponential random variable and a beta random variable. This extends a recent result of Schulte-Geers and Stadje (2017) from Brownian motion with drift to recurrent Bessel processes with drift. Our proof is intuitive and direct while avoiding heavy computations. For this we develop a novel additive decomposition for the square of a Bessel process with drift that may be of independent interest.
Large deviations related to Dynkin--Lamperti arcsine laws for last visit times of Markov processes
Faculty of Arts and Sciences, Kyoto Institute of Technology（京都工艺纤维大学艺术与科学学院）；Department of Mathematics, Graduate School of Science, The University of Osaka（大阪大学理学研究科数学系）；Department of Systems Innovation, Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科系统创新系）
We study the aging behavior of the Random Energy Model (REM) evolving under Metropolis dynamics. We prove that a classical two-time correlation function converges almost surely to the arcsine law distribution function that characterizes activated aging, as predicted in the physics literature, in the optimal domain of the time-scale and temperature parameters where this result can be expected to hold. In the course of the proof we establish that a certain continuous time clock process, after proper rescaling, converges almost surely to a stable subordinator, improving upon the result of Cerny a…
We study the aging behavior of a truncated version of the Random Energy Model evolving under Metropolis dynamics. We prove that the natural time-time correlation function defined through the overlap function converges to an arcsine law distribution function, almost surely in the random environment and in the full range of time scales and temperatures for which such a result can be expected to hold. This establishes that the dynamics ages in the same way as Bouchaud's REM-like trap model, thus extending the universality class of the latter model. The proof relies on a clock process convergence …
Using a method developed by Durrett and Resnick [22] we establish general criteria for the convergence of properly rescaled clock processes of random dynamics in random environments on infinite graphs. This complements the results of [26], [19], and [20]: put together these results provide a unified framework for proving convergence of clock processes. As a first application we prove that Bouchaud's asymmetric trap model on ${\Bbb Z}^d$ exhibits a normal aging behavior for all $d\geq 2$. Namely, we show that certain two-time correlation functions, among which the classical probability to find …
We show analogs of the classical arcsine theorem for the occupation time of a random walk in $(-\infty,0)$ in the case of a small positive drift. To study the asymptotic behavior of the total time spent in $(-\infty,0)$ we consider parametrized classes of random walks, where the convergence of the parameter to zero implies the convergence of the drift to zero. We begin with shift families, generated by a centered random walk by adding to each step a shift constant $a>0$ and then letting $a$ tend to zero. Then we study families of associated distributions. In all cases we arrive at the same lim…
An open problem of interest, first infused into the applied probability community in the work of Bingham and Doney in 1988, (see \cite{Bingham}) is stated as follows: find the distribution of the quadrant occupation time of planar Brownian motion. In this short communication, we study an alternate formulation of this longstanding open problem: let $X(t), Y(t), t \geq 0$ be standard Brownian motions starting at $x,y$ respectively. Find the distribution of the total time $T=Leb\{t \in [0,1]: X(t) \times Y(t) >0\}$, when $x=y=0$, i.e., the occupation time of the union of the first and third quadr…
This article offers a simplified approach to the distribution theory of randomly weighted averages or $P$-means $M_P(X):= \sum_{j} X_j P_j$, for a sequence of i.i.d.random variables $X, X_1, X_2, \ldots$, and independent random weights $P:= (P_j)$ with $P_j \ge 0$ and $\sum_{j} P_j = 1$. The collection of distributions of $M_P(X)$, indexed by distributions of $X$, is shown to encode Kingman's partition structure derived from $P$. For instance, if $X_p$ has Bernoulli$(p)$ distribution on $\{0,1\}$, the $n$th moment of $M_P(X_p)$ is a polynomial function of $p$ which equals the probability gener…
In this paper we investigate the argmin process of Brownian motion $B$ defined by $α_t:=\sup\left\{s \in [0,1]: B_{t+s}=\min_{u \in [0,1]}B_{t+u} \right\}$ for $t \geq 0$. The argmin process $α$ is stationary,with invariant measure which is arcsine distributed. We prove that $(α_t; t \geq 0)$ is a Markov process with the Feller property, and provide its transition kernel $Q_t(x,\cdot)$ for $t>0$ and $x \in [0,1]$. Similar results for the argmin process of random walks and Lévy processes are derived. We also consider Brownian extrema of a given length. We prove that these extrema form a delayed…
Let $(S_k)_{k\ge 1}$ be the classical Bernoulli random walk on the integer line with jump parameters $p\in(0,1)$ and $q=1-p$. The probability distribution of the sojourn time of the walk in the set of non-negative integers up to a fixed time is well-known, but its expression is not simple. By modifying slightly this sojourn time--through a particular counting process of the zeros of the walk as done by Chung & Feller ["On fluctuations in coin-tossings", Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 605-608]-, simpler representations may be obtained for its probability distribution. In the aforementi…
We investigate some simple and surprising properties of a one-dimensional Brownian trajectory with diffusion coefficient $D$ that starts at the origin and reaches $X$ either: (i) at time $T$ or (ii) for the first time at time $T$. We determine the most likely location of the first-passage trajectory from $(0,0)$ to $(X,T)$ and its distribution at any intermediate time $t<T$. A first-passage path typically starts out by being repelled from its final location when $X^2/DT\ll 1$. We also determine the distribution of times when the trajectory first crosses and last crosses an arbitrary intermedia…
