Striking universalities in stochastic resetting processes
Abstract: Given a random process $x(\tau)$ which undergoes stochastic resetting at a constant rate $r$ to a position drawn from a distribution ${\cal P}(x)$, we consider a sequence of dynamical observables $A_1, \dots, A_n$ associated to the intervals between resetting events. We calculate exactly the probabilities of various events related to this sequence: that the last element is larger than all previous ones, that the sequence is monotonically increasing, etc. Remarkably, we find that these probabilities are ``super… Show more
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Cited by 11 publications
References 53 publications
For a one-dimensional Brownian motion starting from the origin, the cumulative distribution of the occupation time V staying above the origin obeys the celebrated arcsine law. In this work, we show how the law is modified for a resetting Brownian motion, where the Brownian is reset to the position x r at random times but with a constant rate r. When x r is exactly equal to zero, we derive the exact expression of the probability distribution P r (V∣0, t) of V during time t, and the moments of V as functions of r and t. P r (V∣0, t) is always symmetric with respect to V = t/2 for arbitrary value of r, but the probability density of V at V = t/2 increases with the increase of r. Interestingly, P r (V∣0, t) at V = t/2 changes from a minimum to a local maximum at a critical value R * ≈ 0.742 338, where R = rt denotes the average number of resetting during time t. Moreover, we consider the case when x r is a random variable and is distributed by a function g(x r ), where g(x r ) is assumed to be symmetric with respect to zero and possesses its maximum at zero. We derive the general expressions of the moments of V when the variance of x r is low. The mean value of V is always equal to t/2, but the fluctuation in x r leads to an increase in the second and third moments of V. Our results provide a quantitative understanding of how stochastic resetting destroys the persistence of Brownian motion.
For a one-dimensional Brownian motion starting from the origin, the cumulative distribution of the occupation time V staying above the origin obeys the celebrated arcsine law. In this work, we show how the law is modified for a resetting Brownian motion, where the Brownian is reset to the position x r at random times but with a constant rate r. When x r is exactly equal to zero, we derive the exact expression of the probability distribution P r (V∣0, t) of V during time t, and the moments of V as functions of r and t. P r (V∣0, t) is always symmetric with respect to V = t/2 for arbitrary value of r, but the probability density of V at V = t/2 increases with the increase of r. Interestingly, P r (V∣0, t) at V = t/2 changes from a minimum to a local maximum at a critical value R * ≈ 0.742 338, where R = rt denotes the average number of resetting during time t. Moreover, we consider the case when x r is a random variable and is distributed by a function g(x r ), where g(x r ) is assumed to be symmetric with respect to zero and possesses its maximum at zero. We derive the general expressions of the moments of V when the variance of x r is low. The mean value of V is always equal to t/2, but the fluctuation in x r leads to an increase in the second and third moments of V. Our results provide a quantitative understanding of how stochastic resetting destroys the persistence of Brownian motion.
Extremal statistics for first-passage trajectories of drifted Brownian motion under stochastic resetting
We study the extreme value statistics of first-passage trajectories generated from a one-dimensional drifted Brownian motion subject to stochastic resetting to the starting point with a constant rate r. Each stochastic trajectory starts from a positive position x 0 and terminates whenever the particle hits the origin for the first time. We obtain an exact expression for the marginal distribution P r ( M | x 0 ) of the maximum displacement M. We find that stochastic resetting has a profound impact on P r ( M | x 0 ) and the expected value ⟨ M ⟩ of M. Depending on the drift velocity v, ⟨ M ⟩ shows three distinct trends of change with r. For v ⩾ 0 , ⟨ M ⟩ decreases monotonically with r, and tends to 2 x 0 as r → ∞ . For v c < v < 0 , ⟨ M ⟩ shows a nonmonotonic dependence on r, in which a minimum ⟨ M ⟩ exists for an intermediate level of r. For v ⩽ v c , ⟨ M ⟩ increases monotonically with r. Moreover, by deriving the propagator and using a path decomposition technique, we obtain, in the Laplace domain, the joint distribution of M and the time tm at which the maximum M is reached. Interestingly, the dependence of the expected value ⟨ t m ⟩ of tm on r is either monotonic or nonmonotonic, depending on the value of v. For v > v m , there is a nonzero resetting rate at which ⟨ t m ⟩ attains its minimum. Otherwise, ⟨ t m ⟩ increases monotonically with r. We provide an analytical determination of two critical values of v, v c ≈ − 1.694 15 D / x 0 and v m ≈ − 1.661 02 D / x 0 , where D is the diffusion constant. Finally, numerical simulations are performed to support our theoretical results.
Extreme value statistics of first-passage trajectories of resetting Brownian motion in an interval
In this work, we consider one-dimensional resetting Brownian motion confined in a unit interval with absorbing boundaries at both ends. The Brownian particle is reset to the starting position x 0 at random times but with a constant rate r, and the process terminates whenever the particle touches any of absorbing boundaries. We study the statistics of the span S, defined as the size of the visited domain during the first-passage process, and the time tm at which the maximum (minimum) displacement is achieved when the particle is absorbed by the left (right) end. We analytically obtain the distribution of S and its mean value. The distribution of S can be significantly altered by the resetting, and the mean value of S can increase, decrease, or remain insignificant as r increases, depending on the value of x 0. Moreover, we find that the mean value ⟨ t m ⟩ has a highly nontrivial dependence on r. When x 0 is close to the boundary, ⟨ t m ⟩ can be reduced by the resetting, and a nonzero resetting rate r = r opt exists for which ⟨ t m ⟩ attains its minimum. Otherwise, ⟨ t m ⟩ has its minimum when the resetting is absent. We observe that the transition occurs at x 0 ≈ 0.180 67 (or x 0 ≈ 0.819 33 ). Such a transition is first-order, characterized by an abrupt jump of r opt from a nonzero value to zero at the transition point.
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