Extremal statistics for a one-dimensional Brownian motion with a reflective boundary

Despite infrequent occurrences, extreme events are of vital importance as they may bring devastating consequences. Examples include natural calamities, such as earthquake, tsunamis and floods, economic collapses, and outbreak of pandemic [1, 2, 3, 4, 5, 6]. Extreme-value statistics (EVS) has been a branch of statistics which deals with the extreme deviations of a random process from its mean behavior. For independent and identically distributed random variables, it is known that the extreme-value distribution falls into three famous universality classes, namely, Gumbel, Fréchet, and Weibull depending on the tails of the distribution of random variables [7]. The study of EVS has also become extremely important in the field of disordered systems [8, 9], fluctuating interfaces [10, 11], interacting spin systems [12], stochastic transport models [13, 14], random matrices [15, 16, 17], epidemic outbreak [18], and computer search algorithms [19, 20, 21].

In recent years, there is an increasing interest in studying the EVS for strongly correlated stochastic processes [22, 23, 24, 25, 26, 27, 28]. We refer the reader to [29, 30] for two recent reviews on the subject. One of the central goals on this subject is to compute the statistics of extremes, i.e., the maximum $M$ of a given trajectory $x(t)$ during an observation time window $\left[0,t\right]$, and the time $t_{m}$ at which the maximum $M$ is reached. A paradigmatic example is a one-dimensional Brownian motion for a fixed duration $t$ starting from the origin, where the joint distribution of $M$ and $t_{m}$ is given by [31]

with the diffusion constant $D$. Marginalization of Eq.(1) over $t_{m}$, one gets the one-sided Gaussian distribution of $M$,

where $\Theta(z)$ is the Heaviside step function such that $\Theta(z)=1$ if $z>0$ and $\Theta(z)=0$ otherwise. In particular, from Eq.(2) one can easily obtain that the expected maximum $M(t)$ of a free Brownian motion grows like $\langle M(t)\rangle=2\sqrt{Dt/\pi}$. On the other hand, one can obtain the marginal distribution of $t_{m}$ by integrating $P_{0}(M,t_{m}|t)$ over $M$,

which is often referred to as the “arcsine law” due to P. Lévy [32, 33, 34]. The name stems from the fact that the cumulative distribution of $t_{m}$ reads $F(z)=\int_{0}^{z}{P_{0}(t_{m}|t)dt_{m}}=(2/\pi)\arcsin\sqrt{z/t}$. A counterintuitive aspect of the U-shaped distribution of Eq.(3) is that its average value $\langle t_{m}\rangle=t/2$ corresponds to the minimum of the distribution, i.e., the less probable outcome, whereas values close to the extrema $t_{m}=0$ and $t_{m}=t$ are much more likely. In fact, the other two times, i.e., the last time $t_{\ell}$ the process crosses the origin and the occupation time $t_{o}$ spent on the positive (or the negative) semi axis, satisfy the same distribution as Eq.(3) as well. The three arcsine laws for Brownian motion play a central role in EVS.

Extreme observables were also studied for variants of Brownian motion. For a class of constrained Brownian motions including Brownian excursions, Brownian meanders, and reflected Brownian bridge, the joint distribution of the maximum $M$ and the extreme time $t_{m}$, and their marginal distributions have been analytically obtained [35]. For the random acceleration process, which is one of the simplest non-Markov stochastic processes, the distribution of $t_{m}$ [36] and the first two moments of the occupation time $t_{o}$ [37] were studied. For the run-and-tumble motion, a representative model of active particles like E. coli, the distributions of three times, i.e., $t_{m}$, $t_{\ell}$ and $t_{o}$ were derived in one dimension [38], and the results were used to compute the statistical properties of the convex hull of a planar run-and-tumble motion [39, 40]. In Refs.[41, 42], the three arcsine laws were generalized to fractional Brownian motion, which is a non-Markovian Gaussian process indexed by the Hurst exponent $H\in\left(0,1\right)$, generalizing standard Brownian motion ($H=\frac{1}{2}$) to account for anomalous diffusion. Using a perturbative expansion in $\epsilon=H-\frac{1}{2}$, the distributions of three times were analytically obtained up to second-order. The distributions of $t_{m}$ and $t_{o}$ are symmetric with respect to half of the duration time $t/2$. In the leading term, the two distributions are the same, but become distinguishable in the sub-leading term. The distribution of $t_{\ell}$ is markedly differently from the other two distributions; especially, it is asymmetric with respect to $t/2$. Recently, Brownian motion and fractional Brownian motion subject to stochastic resetting have received increasing attention due to the emergence of a nonequilibrium steady state and its advantage in accelerating a random search [43, 44, 45, 46]. For a one-dimensional resetting Brownian motion with a fixed duration, the distributions of $M$ and $t_{m}$ have been derived [47, 48], from which statistical properties of the convex hull of a planar resetting Brownian motion were obtained [48]. The distribution of $t_{o}$ for a one-dimensional resetting Brownian motion is considered, and particular attention was paid to the large deviation property of the distribution [49]. Very recently, extreme statistics and spacing distribution for the positions of a one-dimensional $N$ Brownian particles under the simultaneous resetting was investigated [28]. EVS has also been studied for general stochastic processes [50, 51, 52, 53, 31, 54, 55, 56, 57]. Extension to study the distribution of the time difference between the minimum and the maximum for stochastic processes has also been made in [58, 59] and to study the EVS before a first passage time through a specified threshold [60, 61, 62, 63, 64]. The distribution of a related observable, i.e., the number of distinct sites visited by a one-dimensional random walker before hitting a target [65], and the joint distribution of the first-passage time to a target and the number of distinct sites visited [66] have been obtained analytically for several representative Markovian processes, such as simple symmetric random walks, biased random walks, persistent random walks, and resetting random walks. Quite remarkably, the statistics of $t_{m}$ has found applications in convex hull problems [39, 40, 67, 48] and also in detecting whether a stationary process is equilibrium or not [68, 69].

So far, most of previous studies were paid their attentions to the EVS of the stochastic process without confinement. However, in many practical situations, the stochastic process takes place in a confined geometry. For example, the range of animal foraging can get constrained by natural or human-built obstacles, such as a river, mountains, urban areas, roads, and so on. A planar Brownian motion in the presence of an infinite reflective wall was studied [70, 71]. It was shown that the presence of the wall breaks the isotropy of the process and induces a non-trivial effect on the convex hull of the Brownian motion. Very recently, a Brownian particle in a $d$-dimensional ball with radius $R$ with reflecting boundaries was considered, and the statistics of the maximum displacement $M_{x}(t)$ of along the $x$-direction at time $t$ was studied [72]. It was shown the distribution of the fluctuation $(M_{x}(t)-R)/R$ in the long-time limit exhibits a rich variety of behaviors depending on the dimension $d$. In a very recent work [73], the authors studied that EVS of a one-dimensional Brownian motion in the presence of a permeable barrier when starting a position infinitely approaching the barrier. They derived the joint distribution of the maximum displacement $M(t)$ and the time $t_{m}$ at the maximum is reached, from which the marginal distributions of $M(t)$ and $t_{m}$ were also obtained. It was also reported that the confinement produces a profound impact on the first-passage properties of the fractional Brownian motion [74] and the ergodicity of heterogeneous diffusion [75].

In this work, we aim to study the EVS of the Brownian motion in one dimension subject to a reflective boundary at the origin when starting from a positive position $x_{0}$, such that the Brownian motion is confined in the positive half-space. From an intuitive perspective of view, when the total time duration $t$ is much less than the diffusion time $t_{d}=x^{2}_{0}/D$ from the starting position to the reflective boundary, the reflective boundary has not much impact on the Brownian motion and thus the EVS is similar to that of a free Brownian motion. We should emphasize that $t_{d}$ is not confused with the mean first-passage time from $x_{0}$ to the origin. It is well-known that the first-passage time $t_{f}$ follows $P(t_{f})=\frac{1}{\sqrt{4\pi Dt_{f}^{3}}}\exp\left({-{x_{0}^{2}}/{4Dt_{f}}}\right)$. The heavy tail of the distribution leads to the divergence of mean first-passage time. Instead, $t_{d}$ can be understood as the typical value of the first-passage time since one can easily check that $P(t_{f})$ attains its unique maximum at $t_{f}=t_{d}$. A special focus should be paid to the case of $t$ equal to or larger than the order of the magnitude of $t_{d}$ where the reflective boundary comes into effect. We first derive the survival probability of the Brownian particle in the presence of a reflective boundary at the origin and an absorbing boundary at $x=M$, thus leading to the distribution $P(M|x_{0},t)$ of the maximum displacement $M$ within a duration $t$. We also obtain the expectation $\langle M\rangle$ of $M$ as a function of $x_{0}$ and $t$, from which we find that in the long-time limit $\langle M\rangle\sim\sqrt{\pi Dt}$, up to a correction proportional to $x_{0}^{2}/\sqrt{t}$. This is qualitatively the same as the case of the free Brownian motion ($\langle M\rangle\sim 2\sqrt{Dt/\pi}$ for the latter), but with a different prefactor. Moreover, we use a path decomposition technique to obtain the joint distribution $P(M,t_{m}|x_{0},t)$ of the maximum displacement $M$ and the time $t_{m}$ at which $M$ is reached. Marginalization of $P(M,t_{m}|x_{0},t)$ over $M$ to obtain the distribution of $P(t_{m}|x_{0},t)$. For $t>t_{d}$, the distribution of $t_{m}$ deviates from the so-called arcsine law, and is no longer symmetric with respect to $t/2$. Interestingly, we find that the ratio of the expectation of $t_{m}$ to $t$, $\langle t_{m}\rangle/t$, increases monotonically with $t$, and its value approaches 1/2 in the limit $t\ll t_{d}$ and tends towards another constant $(1+2G)/4\approx 0.708$ in the opposite limit $t\gg t_{d}$, where $G$ is the Catalan’s constant. At last, we show how the statistics of the other two times, i.e., the last time $t_{\ell}$ the process passes through the initial position $x_{0}$ and the occupation time $t_{o}$ spent above $x_{0}$, is modified in the presence of the reflective boundary. The expectations of $t_{\ell}$ and $t_{o}$ are analytically obtained. In the short-time and the long-time limits, $\langle t_{\ell}\rangle/t$ tends towards $1/2$. Interestingly, $\langle t_{\ell}\rangle/t$ shows a nonmonotonic dependence on $t$, and a maximum $\langle t_{\ell}\rangle/t$ occurs at a moderate value of $t$. The fraction of the occupation time $\langle t_{o}\rangle/t$ varies monotonically from $1/2$ and approaches to $1$ in the long-time limit.

The rest of the paper is organized as follows. In Sec.II we define the Brownian motion in one dimension subject to a reflective boundary and propose the extreme-value questions we want to study. In Sec.III and Sec.IV, we derive, both in the time domain and in the Laplace domain, the propagator and survival probability of the Brownian motion in an interval with a reflective end and an absorbing end. In Sec.V, we present the marginal distribution of the maximum displacement $M$ and its expected value. Using a path decomposition technique, in Sec.VI we obtain the joint distribution of $M$ and the time $t_{m}$ at which the maximum $M$ is reached. In Sec.VII, we provide the marginal of the extreme time $t_{m}$ and make asymptotic analysis for the expectation of $t_{m}$ in the short-time and the long-time limits. In Sec.VIII, the statistics of the last time visited the starting position and the occupation time spent above the starting position are analytically given. Finally, the main conclusions and perspective are addressed in Sec.IX.

Let us consider a one-dimensional Brownian motion starting at $x_{0}$($>0$) with a reflective boundary at the origin,

where $D$ is the diffusion constant, and ${\xi}(t)$ is a Gaussian white noise with zero mean $\langle{\xi(t)}\rangle=0$ and delta correlator $\langle{\xi(t)}{\xi(t^{\prime})}\rangle=\delta({t-t^{\prime}})$. During a time interval $\left[0,t\right]$, the displacement $x(t)$ of the Brownian reaches its maximum $M$ at time $t_{m}$, see Fig.1 for an illustration. We are interested in the joint distribution $P(M,t_{m}|x_{0},t)$ of $M$ and $t_{m}$, and their marginal distributions of $M$ and $t_{m}$, $P(M|x_{0},t)$ and $P(t_{m}|x_{0},t)$.

We begin with the computation of the probability density function $G(x,t|x_{0})$ of position of the Brownian particle or the propagator in the presence of a reflective wall at $x=0$, when starting from an initial position $x_{0}\in\left[0,M\right)$. On the other hand, the maximum displacement $M$ of the particle within a time duration $\left[0,t\right]$ yields an absorbing boundary at $x=M$.

Differentiating Eq.(27) with respect to $M$ gives the probability density function of $M$,

In Fig.2, we plot the marginal distribution $P(M|x_{0},t)$ for different $t$, but with fixed $x_{0}=1$ and $D=1/2$. For comparison, we have performed simulations that supports our theory. In all simulations, we have used the time step $dt=10^{-5}$ and the results are averaged over $10^{5}$ different realizations. For $t$ much less than the diffusion time $t_{d}=x_{0}^{2}/D$ from the starting position $x_{0}$ to the reflective wall at the origin, the diffusing particle is almost unaffected by reflective boundary such that the distribution $P(M|x_{0},t)$ is similar that of freely diffusive particle shown in Eq.(2), from which we can observe that $P(M|x_{0},t)$ decreases monotonically with $M$. As $t$ increases, the reflective boundary comes into effect, and $P(M|x_{0},t)$ is no longer a monotonic function of $M$. Interestingly, $P(M|x_{0},t)$ can show a unique maximum at an intermediate level of $M$. In the limit $t\gg t_{d}$, the summation in the right-hand side of Eq.(31) is dominated by only one term for $n=0$. In this case, $P(M|x_{0},t)$ decays with $M$ as $M^{-3}$ in the large-$M$ limit.

Taking the derivative of Eq.(30) with respect to $M$, we obtain the marginal distribution $P(M|x_{0},t)$ in the Laplace domain,

It turns out that inverting Eq.(32) is not a trivial task. However, we can conveniently compute the expectation of the maximum displacement $M$ from $\tilde{P}(M|x_{0},s)$, defined as

Performing the Laplace transform for Eq.(33) and utilizing Eq.(32), we have

Prior to the inversion of Eq.(34), we first consider two limiting cases. In the short-time limit ($s\to\infty$) and in the long-time limit ($s\to 0$), Eq.(34) becomes

Performing the inverse Laplace transform for Eq.(37), we obtain

In the short-time limit, $t\ll t_{d}=\frac{x_{0}^{2}}{D}$, the result coincides with the counterpart of the free Brownian motion. In the long-time limit, $t\gg t_{d}$, the expected maximum is weakly related to the initial position $x_{0}$, and grows with $t^{1/2}$ as $t$ increases. Interestingly, the expected maximum under the case is $\pi/2$ times that of the free Brownian motion starting from the origin.

To invert Eq.(34) for the general case, we intend to rewrite Eq.(34) as series expansions. To the end, let us denote by $y=e^{\alpha x_{0}}$, and use the equations $\arctan({1/y})=\sum_{n=0}^{\infty}{\frac{{{{\left({-1}\right)}^{n}}}}{{2n+1}}{y^{-\left({2n+1}\right)}}}$ ($y\geq 1$) and $\cosh(\ln y)=\frac{1+y^{2}}{2y}$ to obtain

Performing the Laplace transform inversion for Eq.(41), we obtain

where ${\rm{erfc}}(x)$ is the complementary error function. We note that the result in Eq.(42) was obtained in a previous work as well (see Eq.(5) in [70]). However, in the present work we additionally obtain the distribution of $M$ and will further derive the statistics of extremal time in the subsequent sections. In Fig.3, we show a log–log plot of $\langle M(x_{0},t)\rangle$ as a function of $t$, where we have fixed $x_{0}=1$ and $D=1/2$. The theoretical prediction is in well agreement with simulations. For comparison, we plot in Fig.3 the expectation $\langle x(t)\rangle$ of the displacement $x(t)$ as a function of $t$ as well. The details of derivation of $\langle x(t)\rangle$ can be found in appendix A. In the short-time limit, $\langle x(t)\rangle=x_{0}$ keeps unchanged as a free Brownian motion, and in the long-time limit $\langle x(t)\rangle\sim\sqrt{t}$, but the prefactor is smaller than that of the expected maximum displacement.

Let us define $P(M,t_{m}|x_{0},t)$ as the joint probability density function that the displacement $x(\tau)$ reaches its maximum $M$ at time $t_{m}$ with a duration $t$, providing that the Brownian starts from the position $x_{0}$ ($>0$). To compute the joint distribution $P(M,t_{m}|x_{0},t)$, we can decompose the trajectory into two parts: a left-hand segment for which $0<\tau<t_{m}$, and a right-hand segment for which $t_{m}<\tau<t$, as shown in Fig.1. The statistical weight of the first segment equals to the propagator $G(M,t_{m}|x_{0})$. However, it turns out that $G(M,t_{m}|x_{0})=0$ which implies that the contribution from this part is zero. To circumvent this problem, we compute $G(M-\epsilon,t_{m}|x_{0})$ and later take the limit $\epsilon\to 0^{+}$ [11]. The statistical weight of the second segment is given by the survival probability $Q(M-\epsilon,t-t_{m}|M)$. Due to the Markov property, the joint probability density $P(M,t_{m}|x_{0},t)$ can be written as the product of the statistical weights of two segments [60],

Here, the normalization factor $\mathcal{N}$ may depend on $D$ and $\epsilon$. In principle, $\mathcal{N}$ depends also on $x_{0}$ and $t$. However, here we assume and verify a posteriori that it is indeed independent of $x_{0}$ and $t$.

Let us integrate $P(M,t_{m}|x_{0},t)$ over $t_{m}$ from 0 to $t$ and over $M$ from $x_{0}$ to $\infty$, and then perform the Laplace transform with respect to $t$, such that the convolution structure of Eq.(43) can be exploited,

Here, $\tilde{G}({M-\epsilon,s|{x_{0}}})$ and $\tilde{Q}({M-\epsilon,s|M})$ can be obtained from Eq.(26) and Eq.(30). Up to the leading order of $\epsilon$, they are

and

where $\alpha=\sqrt{s/D}$ again. By inserting Eq.(45) and Eq.(46) into Eq.(44) and completing the integral over $M$ on the right-hand side of Eq.(44), we obtain the normalization factor $\mathcal{N}$,

Moreover, we perform double Laplace transformations for $P(M,t_{m}|x_{0},t)$ with respect to $t_{m}$ ($\to s$) and $t$ ($\to p$), and use Eq.(45), Eq.(46) and Eq.(47), to obtain

where we have exchanged the order of integration over $t$ and $t_{m}$. By using the inverse Laplace transform relations [77],

where

are the Elliptic theta functions, we obtain $P(M,t_{m}|x_{0},t)$ from Eq.(48),

where $a=M/\sqrt{D}$ and $\nu=x_{0}/\sqrt{D}$. In fact, the joint distribution $P(M,t_{m}|x_{0},t)$ in Eq.(53) can be also obtained directedly from the results in the time domain. This can be done more easily by series expansions of $G(M-\epsilon,t_{m}|x_{0})$ and $Q(M-\epsilon,t-t_{m}|M)$ to the first order in terms of Eq.(8) and Eq.(27), and then inserting them into Eq.(43).

Integrating $P(M,t_{m}|x_{0},t)$ in Eq.(53) over $M$ from $x_{0}$ to $\infty$, we get to the marginal distribution $P(t_{m}|x_{0},t)$ of $t_{m}$ at which the maximum displacement $M$ is achieved,

where