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 set-up. 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 (RW) 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.
We revisit the simple lattice random walk (Pólya walk) and the Sisyphus random walk in $\mathbb{Z}$, in the presence of random restarts. We use a relatively direct approach namely First passage under restart for discrete space and time which was recently developed by us in PRE 103, 052129 (2021) and rederive the first passage properties of these walks under the memoryless geometric restart mechanism. Subsequently, we show how our method could be generalized to arbitrary first passage process subject to more complex restart mechanisms such as sharp, Poisson and Zeta distribution where the latte…
First passage under restart has recently emerged as a conceptual framework suitable for the description of a wide range of phenomena, but the endless variety of ways in which restart mechanisms and first passage processes mix and match hindered the identification of unifying principles and general truths. Hope that these exist came from a recently discovered universality displayed by processes under optimal, constant rate, restart---but extensions and generalizations proved challenging as they marry arbitrarily complex processes and restart mechanisms. To address this challenge, we develop a g…
First passage under restart with branching is proposed as a generalization of first passage under restart. Strong motivation to study this generalization comes from the observation that restart with branching can expedite the completion of processes that cannot be expedited with simple restart; yet a sharp and quantitative formulation of this statement is still lacking. We develop a comprehensive theory of first passage under restart with branching. This reveals that two widely applied measures of statistical dispersion---the coefficient of variation and the Gini index---come together to deter…
Restart -- interrupting a stochastic process followed by a new start -- is known to improve the mean time to its completion, and the general conditions under which such an improvement is achieved are now well understood. Here, we explore how restart affects other important metrics of first-passage phenomena, namely the median and the mode of the first-passage time distribution. Our analysis provides a general criterion for when restart lowers the median time, and demonstrates that restarting is always helpful in reducing the mode. Additionally, we show that simple non-uniform restart strategie…
Stochastic restart may drastically reduce the expected run time of a computer algorithm, expedite the completion of a complex search process, or increase the turnover rate of an enzymatic reaction. These diverse first-passage-time (FPT) processes seem to have very little in common but it is actually quite the other way around. Here we show that the relative standard deviation associated with the FPT of an optimally restarted process, i.e., one that is restarted at a (non-zero) rate which brings the mean FPT to a minimum, is always unity. We interpret, further generalize, and discuss this findi…
Restart is a general framework, of prime importance and wide applicability, for expediting first-passage times and completion times of general stochastic processes. Restart protocols can use either deterministic or stochastic timers. Restart protocols with deterministic timers -- "sharp restart" -- assume a principal role: if there exists a restart protocol that improves mean-performance, then there exists a sharp-restart protocol that performs as good or better. This paper, the first of a duo, presents a comprehensive mean-performance analysis of sharp restart. Using statistical methods, the …
Recently noticed ability of restart to reduce the expected completion time of first-passage processes allows appealing opportunities for performance improvement in a variety of settings. However, complex stochastic processes often exhibit several possible scenarios of completion which are not equally desirable in terms of efficiency. Here we show that restart may have profound consequences on the splitting probabilities of a Bernoulli-like first-passage process, i.e. of a process which can end with one of two outcomes. Particularly intriguing in this respect is the class of problems where a ca…
We develop a Landau like theory to characterize the phase transitions in resetting systems. Restart can either accelerate or hinder the completion of a first passage process. The transition between these two phases is characterized by the behavioral change in the order parameter of the system namely the optimal restart rate. Like in the original theory of Landau, the optimal restart rate can undergo a first or second order transition depending on the details of the system. Nonetheless, there exists no unified framework which can capture the onset of such novel phenomena. We unravel this in a c…
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 upon 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, …
First-passage processes can be divided in two classes: those that are accelerated by the introduction of restart and those that display an opposite response. In physical systems, a transition between the two classes may occur as governing parameters are varied to cross a universal tipping point. However, a fully tractable model system to teach us how this transition unfolds is still lacking. To bridge this gap, we quantify the effect of stochastic restart on the first-passage time of a drift-diffusion process to an absorbing boundary. There, we find that the transition is governed by the Pécle…
Classical first-passage times under restart are used in a wide variety of models, yet the quantum version of the problem still misses key concepts. We study the quantum hitting time with restart using a monitored quantum walk. The restart strategy eliminates the problem of dark states, i.e. cases where the particle evades detection, while maintaining the ballistic propagation which is important for fast search. We find profound effects of quantum oscillations on the restart problem, namely a type of instability of the mean detection time, and optimal restart times that form staircases, with su…
The escape of the randomly accelerated undamped particle from the finite interval under action of stochastic resetting is studied. The motion of such a particle is described by the full Langevin equation and the particle is characterized by the position and velocity. We compare three resetting protocols, which restarts velocity or position (partial resetting) and the whole motion (position and velocity -- full resetting). Using the mean first passage time we assess efficiency of restarting protocols in facilitating or suppressing the escape kinetics. There are fundamental differences between p…
Biased lattice random walks (BLRW) are used to model random motion with drift in a variety of empirical situations in engineering and natural systems such as phototaxis, chemotaxis or gravitaxis. When motion is also affected by the presence of external borders resulting from natural barriers or experimental apparatuses, modelling biased random movement in confinement becomes necessary. To study these scenarios, confined BLRW models have been employed but so far only through computational techniques due to the lack of an analytic framework. Here, we lay the groundwork for such an analytical app…
We develop a model to compute the first-passage time of a random walker in a crowded environment. Hard-core particles with the same size and diffusion coefficient than the tracer diffuse, and the model allows to compute the first passage time of the tracer on euclidian lattices. The result is compared to classical Nakazato-Kitahara model, and extends previous results obtained for persistent random walker. The crowding in a confined media acts as a memory effect, and thus lead to a persistent-like behavior.
We study the survival probability and the corresponding first passage time density of fractional Brownian motion confined to a two-dimensional open wedge domain with absorbing boundaries. By analytical arguments and numerical simulation we show that in the long time limit the first passage time density scales as t**{-1+pi*(2H-2)/(2*Theta)} in terms of the Hurst exponent H and the wedge angle Theta. We discuss this scaling behaviour in connection with the reaction kinetics of FBM particles in a one-dimensional domain.
In this paper we consider a particular version of the random walk with restarts: random reset events which bring suddenly the system to the starting value. We analyze its relevant statistical properties like the transition probability and show how an equilibrium state appears. Formulas for the first-passage time, high-water marks and other extreme statistics are also derived: we consider counting problems associated naturally to the system. Finally we indicate feasible generalizations useful for interpreting different physical effects.
We study the first-passage time to the origin of a mortal Brownian particle, with mortality rate $ μ$, diffusing in one dimension. The particle starts its motion from $ x>0 $ and it is subject to stochastic resetting with constant rate $ r $. We first unveil the relation between the probability of reaching the target and the mean first-passage time of the corresponding problem in absence of mortality, which allows us to deduce under which conditions the former can be increased by adjusting the restart rate. We then consider the first-passage time conditioned on the event that the particle reac…
The first passage search of a diffusing target (prey) by multiple searchers (predators) in confinement is an important problem in the stochastic process literature. While the analogous problem in open space has been studied in some details, a systematic study in confined space is still lacking. In this paper, we study the first passage times for this problem in 1,2 and $3-$dimensions. Due to confinement, the survival probability of the target takes a form $\sim e^{-t/τ}$ at large times $t$. The characteristic capture timescale $τ$ associated with the rare capture events are rather challenging …
Mirror transitions in diffusion with stochastic resetting confined on a ring
Universidad Autónoma Metropolitana（墨西哥自治 metropolitan 大学）；Universidad Autónoma de Chiapas（恰帕斯自治大学）；Instituto de Física, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理研究所）
Resetting or restart, when applied to a stochastic process, usually brings its dynamics to a time-independent stationary state. In turn, the optimal resetting rate makes the mean time to reach a target to be the shortest one. These and other intriguing problems have been intensively studied in the case of ordinary diffusive processes over the last decade. In this paper we consider the influence of stochastic resetting on a diffusive motion modeled in terms of the non-linear differential equation. The reason for its non-linearity is the power-law dependence of the diffusion coefficient on the p…
We consider an anisotropic needle-like Brownian particle with nematic symmetry confined in a $2D$ domain. For this system, the coupling of translational and rotational diffusion makes the process ${\bf x} (t)$ of the positions of the particle non Markovian. Using scaling arguments, a Gaussian approximation and numerical methods, we determine the mean first passage time $\langle\mathbf{T}\rangle$ of the particle to a target of radius $a$ and show in particular that $\langle\mathbf{T}\rangle \sim a^{-1/2}$ for $a\to 0$, in contrast with the classical logarithmic divergence obtained in the case o…
We survey recent results on first-passage processes in unbounded cones and their applications to ordering of particles undergoing Brownian motion in one dimension. We first discuss the survival probability S(t) that a diffusing particle, in arbitrary spatial dimension, remains inside a conical domain up to time t. In general, this quantity decays algebraically S ~ t^{-beta} in the long-time limit. The exponent beta depends on the opening angle of the cone and the spatial dimension, and it is root of a transcendental equation involving the associated Legendre functions. The exponent becomes a f…
Self-interacting random walks are endowed with long range memory effects that emerge from the interaction of the random walker at time $t$ with the territory that it has visited at earlier times $t'<t$. This class of non Markovian random walks has applications in a broad range of examples, ranging from insects to living cells, where a random walker modifies locally its environment -- leaving behind footprints along its path, and in turn responds to its own footprints. Because of their inherent non Markovian nature, the exploration properties of self-interacting random walks have remained elusi…
Aging, the dependence of the dynamics of a physical process on the time $t_a$ since its original preparation, is observed in systems ranging from the motion of charge carriers in amorphous semiconductors over the blinking dynamics of quantum dots to the tracer dispersion in living biological cells. Here we study the effects of aging on one of the most fundamental properties of a stochastic process, the first passage dynamics. We find that for an aging continuous time random walk process the scaling exponent of the density of first passage times changes twice as the aging progresses and reveals…
Non-equilibrium and first-passage properties of generalized Ornstein-Uhlenbeck processes under stochastic resetting
Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics（莫斯科电子与数学研究所，国立高等经济大学）；Chemistry Department, M.V.Lomonosov Moscow State University（莫斯科大学化学系）
For a random walk on a confined one-dimensional domain, we consider mean first passage times (MFPT) in the presence of a mobile trap. The question we address is whether a mobile trap can improve capture times over a stationary trap. We consider two scenarios: a randomly moving trap and an oscillating trap. In both cases, we find that a stationary trap actually performs better (in terms of reducing expected capture time) than a very slowly moving trap; however, a trap moving sufficiently fast performs better than a stationary trap. We explicitly compute the thresholds that separate the two regi…
We analyze velocity-jump process models of persistent search for a single target on a bounded domain. The searcher proceeds along ballistic trajectories and is absorbed upon collision with the target boundary. When reaching the domain boundary, the searcher chooses a random direction for its new trajectory. For circular domains and targets, we can approximate the mean first passage time (MFPT) using a Markov chain approximation of the search process. Our analysis and numerical simulations reveal that the time to find the target decreases for targets closer to the domain boundary. When there is…
Extreme value functionals of stochastic processes are inverse functionals of the first passage time -- a connection that renders their probability distribution functions equivalent. Here, we deepen this link and establish a framework for analyzing extreme value statistics of ergodic reversible Markov processes in confining potentials on the hand of the underlying relaxation eigenspectra. We derive a chain of inequalities, which bounds the long-time asymptotics of first passage densities, and thereby extrema, from above and from below. The bounds involve a time integral of the transition probab…
In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as…
The first-passage time (FPT), defined as the time a random walker takes to reach a target point in a confining domain, is a key quantity in the theory of stochastic processes. Its importance comes from its crucial role to quantify the efficiency of processes as varied as diffusion-limited reactions, target search processes or spreading of diseases. Most methods to determine the FPT properties in confined domains have been limited to Markovian (memoryless) processes. However, as soon as the random walker interacts with its environment, memory effects can not be neglected. Examples of non Markov…
We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function $ψ(t) \simeq t^{-1-α}$) ($0\le α\le 2$). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time $t_a$, the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs describe systems su…
We study the lattice random walk dynamics in a heterogeneous space of two media separated by an interface and having different diffusivity and bias. Depending on the position of the interface, there exist two exclusive ways to model the dynamics: (1) Type A dynamics whereby the interface is placed between two lattice points, and (2) Type B dynamics whereby the interface is placed on a lattice point. For both types, we obtain exact results for the one-dimensional generating function of the Green's function or propagator for the composite system in unbounded domain as well as domains confined wi…
