First Passage under Restart with Branching

Continuous gated first-passage processes

Gated first-passage processes, where completion depends on both hitting a target and satisfying additional constraints, are prevalent across various fields. Despite their significance, analytical solutions to basic problems remain unknown, e.g. the detection time of a diffusing particle by a gated interval, disk, or sphere. In this paper, we elucidate the challenges posed by continuous gated first-passage processes and present a renewal framework to overcome them. This framework offers a unified approach for a wide range of problems, including those with single-point, half-line, and interval targets. The latter have so far evaded exact solutions. Our analysis reveals that solutions to gated problems can be obtained directly from the ungated dynamics. This, in turn, reveals universal properties and asymptotic behaviors, shedding light on cryptic intermediate-time regimes and refining the notion of high-crypticity for continuous-space gated processes. Moreover, we extend our formalism to higher dimensions, showcasing its versatility and applicability. Overall, this work provides valuable insights into the dynamics of continuous gated first-passage processes and offers analytical tools for studying them across diverse domains.

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime

We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

Thermodynamic trade-off relation for first passage time in resetting processes

Resetting is a strategy for boosting the speed of a target-searching process. Since its introduction over a decade ago, most studies have been carried out under the assumption that resetting takes place instantaneously. However, due to its irreversible nature, resetting processes incur a thermodynamic cost, which becomes infinite in the case of instantaneous resetting. Here, we take into consideration both the cost and the first passage time (FPT) required for a resetting process, in which the reset or return to the initial location is implemented using a trapping potential over a finite but random time period. An iterative generating function and a counting functional method à la Feynman and Kac are employed to calculate the FPT and the average work for this process. From these results, we obtain an explicit form of the time-cost trade-off relation, which provides the lower bound of the mean FPT for a given work input when the trapping potential is linear. This trade-off relation clearly shows that instantaneous resetting is achievable only when an infinite amount of work is provided. More surprisingly, the trade-off relation derived from the linear potential seems to be valid for a wide range of trapping potentials. In addition, we have also shown that the fixed-time or sharp resetting can further enhance the trade-off relation compared to that of the stochastic resetting.

Diffusion with partial resetting

Inspired by many examples in nature, stochastic resetting of random processes has been studied extensively in the past decade. In particular, various models of stochastic particle motion were considered where, upon resetting, the particle is returned to its initial position. Here we generalize the model of diffusion with resetting to account for situations where a particle is returned only a fraction of its distance to the origin, e.g., half way. We show that this model always attains a steady-state distribution which can be written as an infinite sum of independent, but not identical, Laplace random variables. As a result, we find that the steady-state transitions from the known Laplace form which is obtained in the limit of full resetting to a Gaussian form, which is obtained close to the limit of no resetting. A similar transition is shown to be displayed by drift diffusion whose steady state can also be expressed as an infinite sum of independent random variables. Finally, we extend our analysis to capture the temporal evolution of drift diffusion with partial resetting, providing a bottom-up probabilistic construction that yields a closed-form solution for the time-dependent distribution of this process in Fourier-Laplace space. Possible extensions and applications of diffusion with partial resetting are discussed.

Time-averaging and nonergodicity of reset geometric Brownian motion with drift

How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Δ)≠TAMSD(Δ) and Variance(Δ)≠TAMSD(Δ) at short lag times Δ and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Δ/T≪1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.

First passage of a diffusing particle under stochastic resetting in bounded domains with spherical symmetry

We investigate the first passage properties of a Brownian particle diffusing freely inside a d-dimensional sphere with absorbing spherical surface subject to stochastic resetting. We derive the mean time to absorption (MTA) as functions of resetting rate γ and initial distance r of the particle to the center of the sphere. We find that when r>r_{c} there exists a nonzero optimal resetting rate γ_{opt} at which the MTA is a minimum, where r_{c}=sqrt[d/(d+4)]R and R is the radius of the sphere. As r increases, γ_{opt} exhibits a continuous transition from zero to nonzero at r=r_{c}. Furthermore, we consider that the particle lies between two two-dimensional or three-dimensional concentric spheres with absorbing boundaries, and obtain the domain in which resetting expedites the MTA, which is (R_{1},r_{c_{1}})∪(r_{c_{2}},R_{2}), with R_{1} and R_{2} being the radii of inner and outer spheres, respectively. Interestingly, when R_{1}/R_{2} is less than a critical value, γ_{opt} exhibits a discontinuous transition at r=r_{c_{1}}; otherwise, such a transition is continuous. However, at r=r_{c_{2}} the transition is always continuous.

Geometric Brownian motion under stochastic resetting: A stationary yet nonergodic process

We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a canonical stochastic multiplicative process for nonstationary and nonergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains nonergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the nonergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable state, and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand-alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.

Unified Approach to Gated Reactions on Networks

For two molecules to react they first have to meet. Yet, reaction times are rarely on par with the first-passage times that govern such molecular encounters. A prime reason for this discrepancy is stochastic transitions between reactive and nonreactive molecular states, which results in effective gating of product formation and altered reaction kinetics. To better understand this phenomenon we develop a unifying approach to gated reactions on networks. We first show that the mean and distribution of the gated reaction time can always be expressed in terms of ungated first-passage and return times. This relation between gated and ungated kinetics is then explored to reveal universal features of gated reactions. The latter are exemplified using a diverse set of case studies which are also used to expose the exotic kinetics that arises due to molecular gating.

A Semi-Deterministic Random Walk with Resetting

We consider a discrete-time random walk (xt) which, at random times, is reset to the starting position and performs a deterministic motion between them. We show that the quantity Prxt+1=n+1|xt=n,n→∞ determines if the system is averse, neutral or inclined towards resetting. It also classifies the stationary distribution. Double barrier probabilities, first passage times and the distribution of the escape time from intervals are determined.

First passage under restart for discrete space and time: Application to one-dimensional confined lattice random walks

First passage under restart has recently emerged as a conceptual framework to study various stochastic processes under restart mechanism. Emanating from the canonical diffusion problem by Evans and Majumdar, restart has been shown to outperform the completion of many first-passage processes which otherwise would take longer time to finish. However, most of the studies so far assumed continuous time underlying first-passage time processes and moreover considered continuous time resetting restricting out restart processes broken up into synchronized time steps. To bridge this gap, in this paper, we study discrete space and time first-passage processes under discrete time resetting in a general setup without specifying their forms. We sketch out the steps to compute the moments and the probability density function which is often intractable in the continuous time restarted process. A criterion that dictates when restart remains beneficial is then derived. We apply our results to a symmetric and a biased random walker in one-dimensional lattice confined within two absorbing boundaries. Numerical simulations are found to be in excellent agreement with the theoretical results. Our method can be useful to understand the effect of restart on the spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions.

Extremal statistics for stochastic resetting systems

While averages and typical fluctuations often play a major role in understanding the behavior of a nonequilibrium system, this nonetheless is not always true. Rare events and large fluctuations are also pivotal when a thorough analysis of the system is being done. In this context, the statistics of extreme fluctuations in contrast to the average plays an important role, as has been discussed in fields ranging from statistical and mathematical physics to climate, finance, and ecology. Herein, we study extreme value statistics (EVS) of stochastic resetting systems, which have recently gained significant interest due to its ubiquitous and enriching applications in physics, chemistry, queuing theory, search processes, and computer science. We present a detailed analysis for the finite and large time statistics of extremals (maximum and arg-maximum, i.e., the time when the maximum is reached) of the spatial displacement in such system. In particular, we derive an exact renewal formula that relates the joint distribution of maximum and arg-maximum of the reset process to the statistical measures of the underlying process. Benchmarking our results for the maximum of a reset trajectory that pertain to the Gumbel class for large sample size, we show that the arg-maximum density attains a uniform distribution independent of the underlying process at a large observation time. This emerges as a manifestation of the renewal property of the resetting mechanism. The results are augmented with a wide spectrum of Markov and non-Markov stochastic processes under resetting, namely, simple diffusion, diffusion with drift, Ornstein-Uhlenbeck process, and random acceleration process in one dimension. Rigorous results are presented for the first two setups, while the latter two are supported with heuristic and numerical analysis.

Mitigating long transient time in deterministic systems by resetting

How long does a trajectory take to reach a stable equilibrium point in the basin of attraction of a dynamical system? This is a question of quite general interest and has stimulated a lot of activities in dynamical and stochastic systems where the metric of this estimation is often known as the transient or first passage time. In nonlinear systems, one often experiences long transients due to their underlying dynamics. We apply resetting or restart, an emerging concept in statistical physics and stochastic process, to mitigate the detrimental effects of prolonged transients in deterministic dynamical systems. We show that resetting the intrinsic dynamics intermittently to a spatial control line that passes through the equilibrium point can dramatically expedite its completion, resulting in a huge reduction in mean transient time and fluctuations around it. Moreover, our study reveals the emergence of an optimal restart time that globally minimizes the mean transient time. We corroborate the results with detailed numerical studies on two canonical setups in deterministic dynamical systems, namely, the Stuart-Landau oscillator and the Lorenz system. The key features-expedition of transient time-are found to be very generic under different resetting strategies. Our analysis opens up a door to control the mean and fluctuations in transient time by unifying the original dynamics with an external stochastic or periodic timer and poses open questions on the optimal way to harness transients in dynamical systems.

Asymmetric stochastic resetting: Modeling catastrophic events

In the classical stochastic resetting problem, a particle, moving according to some stochastic dynamics, undergoes random interruptions that bring it to a selected domain, and then the process recommences. Hitherto, the resetting mechanism has been introduced as a symmetric reset about the preferred location. However, in nature, there are several instances where a system can only reset from certain directions, e.g., catastrophic events. Motivated by this, we consider a continuous stochastic process on the positive real line. The process is interrupted at random times occurring at a constant rate, and then the former relocates to a value only if the current one exceeds a threshold; otherwise, it follows the trajectory defined by the underlying process without resetting. An approach to obtain the exact nonequilibrium steady state of such systems and the mean first passage time to reach the origin is presented. Furthermore, we obtain the explicit solutions for two different model systems. Some of the classical results found in symmetric resetting, such as the existence of an optimal resetting, are strongly modified. Finally, numerical simulations have been performed to verify the analytical findings, showing an excellent agreement.

Experimental Realization of Diffusion with Stochastic Resetting

Stochastic resetting is prevalent in natural and man-made systems, giving rise to a long series of nonequilibrium phenomena. Diffusion with stochastic resetting serves as a paradigmatic model to study these phenomena, but the lack of a well-controlled platform by which this process can be studied experimentally has been a major impediment to research in the field. Here, we report the experimental realization of colloidal particle diffusion and resetting via holographic optical tweezers. We provide the first experimental corroboration of central theoretical results and go on to measure the energetic cost of resetting in steady-state and first-passage scenarios. In both cases, we show that this cost cannot be made arbitrarily small because of fundamental constraints on realistic resetting protocols. The methods developed herein open the door to future experimental study of resetting phenomena beyond diffusion.

Work Fluctuations and Jarzynski Equality in Stochastic Resetting

We consider the paradigm of an overdamped Brownian particle in a potential well, which is modulated through an external protocol, in the presence of stochastic resetting. Thus, in addition to the short range diffusive motion, the particle also experiences intermittent long jumps that reset the particle back at a preferred location. Due to the modulation of the trap, work is done on the system and we investigate the statistical properties of the work fluctuations. We find that the distribution function of the work typically, in asymptotic times, converges to a universal Gaussian form for any protocol as long as that is also renewed after each resetting event. When observed for a finite time, we show that the system does not generically obey the Jarzynski equality that connects the finite time work fluctuations to the difference in free energy. Nonetheless, we identify herein a restricted set of protocols which embraces the relation. In stark contrast, the Jarzynski equality is always fulfilled when the protocols continue to evolve without being reset. We present a set of exactly solvable models, demonstrate the validation of our theory and carry out numerical simulations to illustrate these findings. Finally, we have pointed out possible realistic implementations for resetting in experiments using the so-called engineered swift equilibration.

Kinetics of contact processes under segregation

The kinetics of contact processes are determined by the interplay among local mass transfer mechanisms, spatial heterogeneity, and segregation. Determining the macroscopic behavior of a wide variety of phenomena across the disciplines requires linking reaction times to the statistical properties of spatially fluctuating quantities. We formulate the dynamics of advected agents interacting with segregated immobile components in terms of a chemical continuous-time random walk. The inter-reaction times result from the first-passage times of mobile species to and across reactive regions, and available immobile reactants undergo a restart procedure. Segregation leads to memory effects and enhances the role of concentration fluctuations in the large-scale dynamics.

Time-dependent density of diffusion with stochastic resetting is invariant to return speed

The canonical Evans-Majumdar model for diffusion with stochastic resetting to the origin assumes that resetting takes zero time: upon resetting the diffusing particle is teleported back to the origin to start its motion anew. However, in reality getting from one place to another takes a finite amount of time which must be accounted for as diffusion with resetting already serves as a model for a myriad of processes in physics and beyond. Here we consider a situation where upon resetting the diffusing particle returns to the origin at a finite (rather than infinite) speed. This creates a coupling between the particle's random position at the moment of resetting and its return time, and further gives rise to a nontrivial cross-talk between two separate phases of motion: the diffusive phase and the return phase. We show that each of these phases relaxes to the steady state in a unique manner; and while this could have also rendered the total relaxation dynamics extremely nontrivial, our analysis surprisingly reveals otherwise. Indeed, the time-dependent distribution describing the particle's position in our model is completely invariant to the speed of return. Thus, whether returns are slow or fast, we always recover the result originally obtained for diffusion with instantaneous returns to the origin.

Nonrenewal resetting of scaled Brownian motion

We investigate an intermittent stochastic process in which diffusive motion with a time-dependent diffusion coefficient, D(t)∼t^{α-1}, α>0 (scaled Brownian motion), is stochastically reset to its initial position and starts anew. The resetting follows a renewal process with either an exponential or a power-law distribution of the waiting times between successive renewals. The resetting events, however, do not affect the time dependence of the diffusion coefficient, so that the whole process appears to be a nonrenewal one. We discuss the mean squared displacement of a particle and the probability density function of its positions in this process. We show that scaled Brownian motion with resetting demonstrates rich behavior whose properties essentially depend on the interplay of the parameters of the resetting process and the particle's displacement infree motion. The motion of particles can remain almost unaffected by resetting but can also get slowed down or even be completely suppressed. Especially interesting are the nonstationary situations in which the mean squared displacement stagnates but the distribution of positions does not tend to any steady state. This behavior is compared to the situation [discussed in the companion paper; A. S. Bodrova et al., Phys. Rev. E 100, 012120 (2019)10.1103/PhysRevE.100.012120] in which the memory of the value of the diffusion coefficient at a resetting time is erased, so that the whole process is a fully renewal one. We show that the properties of the probability densities in such processes (erasing or retaining the memory on the diffusion coefficient) are vastly different.

First passage under stochastic resetting in an interval

We consider a Brownian particle diffusing in a one-dimensional interval with absorbing end points. We study the ramifications when such motion is interrupted and restarted from the same initial configuration. We provide a comprehensive study of the first-passage properties of this trapping phenomena. We compute the mean first-passage time and derive the criterion on which restart always expedites the underlying completion. We show how this set-up is a manifestation of a success-failure problem. We obtain the success and failure rates and relate them with the splitting probabilities, namely the probability that the particle will eventually be trapped on either of the boundaries without hitting the other one. Numerical studies are presented to support our analytic results.

First passage of a particle in a potential under stochastic resetting: A vanishing transition of optimal resetting rate

First passage in a stochastic process may be influenced by the presence of an external confining potential, as well as "stochastic resetting" in which the process is repeatedly reset back to its initial position. Here, we study the interplay between these two strategies, for a diffusing particle in a one-dimensional trapping potential V(x), being randomly reset at a constant rate r. Stochastic resetting has been of great interest as it is known to provide an "optimal rate" (r_{*}) at which the mean first passage time is a minimum. On the other hand, an attractive potential also assists in the first capture process. Interestingly, we find that for a sufficiently strong external potential, the advantageous optimal resetting rate vanishes (i.e., r_{*}→0). We derive a condition for this optimal resetting rate vanishing transition, which is continuous. We study this problem for various functional forms of V(x), some analytically, and the rest numerically. We find that the optimal rate r_{*} vanishes with a deviation from the critical strength of the potential as a power law with an exponent β which appears to be universal.