Transport properties of random walks under stochastic noninstantaneous resetting

Optimal conditions for first passage of jump processes with resetting

We investigate the first passage time beyond a barrier located at b≥0 of a random walk with independent and identically distributed jumps, starting from x0=0. The walk is subject to stochastic resetting, meaning that after each step the evolution is restarted with fixed probability r. We consider a resetting protocol that is an intermediate situation between a random walk (r=0) and an uncorrelated sequence of jumps all starting from the origin (r=1) and derive a general condition for determining when restarting the process with 0 Collapse

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Affiliation(s)

Mattia Radice Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
Giampaolo Cristadoro Dipartimento di Matematica e Applicazioni, Università degli Studi Milano-Bicocca, 20126 Milan, Italy
Samudrajit Thapa Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

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Granular gases under resetting

We investigate the granular temperatures in force-free granular gases under exponential resetting. When a resetting event occurs, the granular temperature attains its initial value, whereas it decreases because of the inelastic collisions between the resetting events. We develop a theory and perform computer simulations for granular gas cooling in the presence of Poissonian resetting events. We also investigate the probability density function to quantify the distribution of granular temperatures. Our theory may help us to understand the behavior of nonperiodically driven granular systems.

Drift-diffusive resetting search process with stochastic returns: Speedup beyond optimal instantaneous return

Stochastic resetting has recently emerged as a proficient strategy to reduce the completion time for a broad class of first-passage processes. In the canonical setup, one intermittently resets a given system to its initial configuration only to start afresh and continue evolving in time until the target goal is met. This is, however, an instantaneous process and thus less feasible for any practical purposes. A crucial generalization in this regard is to consider a finite-time return process which has significant ramifications to the firstpassage properties. Intriguingly, it has recently been shown that for diffusive search processes, returning in finite but stochastic time can gain significant speedup over the instantaneous resetting process. Unlike diffusion which has a diverging mean completion time, in this paper, we ask whether this phenomena can also be observed for a first-passage process with finite mean completion time. To this end, we explore the setup of a classical drift-diffusive search process in one dimension with stochastic resetting and further assume that the return phase is modulated by a potential U(x)=λ|x| with λ>0. For this process, we compute the mean first-passage time exactly and underpin its characteristics with respect to the resetting rate and potential strength. We find a unified phase space that allows us to explore and identify the system parameter regions where stochastic return supersedes over both the underlying process and the process under instantaneous resetting. Furthermore and quite interestingly, we find that for a range of parameters the mean completion time under stochastic return protocol can be reduced further than the optimally restarted instantaneous processes. We thus believe that resetting with stochastic returns can serve as a better optimization strategy owing to its dominance over classical first passage under resetting.

Search with stochastic home returns can expedite classical first passage under resetting

Classical first passage under resetting is a paradigm in the search process. Despite its multitude of applications across interdisciplinary sciences, experimental realizations of such resetting processes posit practical challenges in calibrating these zero time irreversible transitions. Here, we consider a strategy in which resetting is performed using finite-time return protocols in lieu of instantaneous returns. These controls could also be accompanied with random fluctuations or errors allowing target detection even during the return phase. To better understand the phenomena, we develop a unified renewal approach that can encapsulate arbitrary search processes centered around home in a fairly general topography containing targets, various resetting times, and return mechanisms in arbitrary dimensions. While such finite-time protocols would apparently seem to prolong the overall search time in comparison to the instantaneous resetting process, we show on the contrary that a significant speed-up can be gained by leveraging the stochasticity in home returns. The formalism is then explored to reveal a universal criterion distilling the benefits of this strategy. We demonstrate how this general principle can be utilized to improve overall performance of a one-dimensional diffusive search process reinforced with experimentally feasible parameters. We believe that such strategies designed with inherent randomness can be made optimal with precise controllability in complex search processes.

Fractional heterogeneous telegraph processes: Interplay between heterogeneity, memory, and stochastic resetting

Fractional heterogeneous telegraph processes are considered in the framework of telegrapher's equations accompanied by memory effects. The integral decomposition method is developed for the rigorous treating of the problem. Exact solutions for the probability density functions and the mean squared displacements are obtained. A relation between the fractional heterogeneous telegrapher's equation and the corresponding Langevin equation has been established in the framework of the developed subordination approach. The telegraph process in the presence of stochastic resetting has been studied, as well. An exact expression for both the nonequilibrium stationary distributions/states and the mean squared displacements are obtained.

Random walks with spatial and temporal resets can explain individual and colony-level searching patterns in ants

Central place foragers, such as many ants, exploit the environment around their nest. The extent of their foraging range is a function of individual movement, but how the movement patterns of large numbers of foragers result in an emergent colony foraging range remains unclear. Here, we introduce a random walk model with stochastic resetting to depict the movements of searching ants. Stochastic resetting refers to spatially resetting at random times the position of agents to a given location, here the nest of searching ants. We investigate the effect of a range of resetting mechanisms and compare the macroscopic predictions of our model to laboratory and field data. We find that all returning mechanisms very robustly ensure that scouts exploring the surroundings of a nest will be exponentially distributed with distance from the nest. We also find that a decreasing probability for searching ants to return to their nest is compatible with empirical data, resulting in scouts going further away from the nest as the number of foraging trips increases. Our findings highlight the importance of resetting random walk models for depicting the movements of central place foragers and nurture novel questions regarding the searching behaviour of ants.

Kuramoto model subject to subsystem resetting: How resetting a part of the system may synchronize the whole of it

We introduce and investigate the effects of a new class of stochastic resetting protocol called subsystem resetting, whereby a subset of the system constituents in a many-body interacting system undergoes bare evolution interspersed with simultaneous resets at random times, while the remaining constituents evolve solely under the bare dynamics. Here, by reset is meant a reinitialization of the dynamics from a given state. We pursue our investigation within the ambit of the well-known Kuramoto model of coupled phase-only oscillators of distributed natural frequencies. Here, the reset protocol corresponds to a chosen set of oscillators being reset to a synchronized state at random times. We find that the mean ω_{0} of the natural frequencies plays a defining role in determining the long-time state of the system. For ω_{0}=0, the system reaches a synchronized stationary state at long times, characterized by a time-independent nonzero value of the synchronization order parameter that quantifies macroscopic order in the system. Moreover, we find that resetting even an infinitesimal fraction of the total number of oscillators, in the extreme limit of infinite resetting rate, has the drastic effect of synchronizing the entire system, even in parameter regimes in which the bare evolution does not support global synchrony. By contrast, for ω_{0}≠0, the dynamics allows at long times either a synchronized stationary state or an oscillatory synchronized state, with the latter characterized by an oscillatory behavior as a function of time of the order parameter, with a nonzero time-independent time average. Our results thus imply that the nonreset subsystem always gets synchronized at long times through the act of resetting of the reset subsystem. Our results, analytical using the Ott-Antonsen ansatz as well as those based on numerical simulations, are obtained for two representative oscillator frequency distributions, namely, a Lorentzian and a Gaussian. Given that it is easier to reset a fraction of the system constituents than the entire system, we discuss how subsystem resetting may be employed as an efficient mechanism to control attainment of global synchrony in the Kuramoto system.

Intermittent random walks under stochastic resetting

We analyze a one-dimensional intermittent random walk on an unbounded domain in the presence of stochastic resetting. In this process, the walker alternates between local intensive search, diffusion, and rapid ballistic relocations in which it does not react to the target. We demonstrate that Poissonian resetting leads to the existence of a non-equilibrium steady state. We calculate the distribution of the first arrival time to a target along with its mean and show the existence of an optimal reset rate. In particular, we prove that the initial condition of the walker, i.e., either starting diffusely or relocating, can significantly affect the long-time properties of the search process. Moreover, we demonstrate the presence of distinct parameter regimes for the global optimization of the mean first arrival time when ballistic and diffusive movements are in direct competition.

Truncated stochastically switching processes

There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation (hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.

Random Walks on Comb-like Structures under Stochastic Resetting

We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker's motion along the backbone is diffusive and it performs short jumps separated by random resting periods along fingers. We take into account two different types of resetting acting separately: global resetting from any point in the comb to the initial position and resetting from a finger to the corresponding backbone. We analyze the interplay between the waiting process and Markovian and non-Markovian resetting processes on the overall mean squared displacement. The Markovian resetting from the fingers is found to induce normal diffusion, thereby minimizing the trapping effect of fingers. In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, with two of them subdiffusive and one of them diffusive. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. As for global resetting, its effect is even more drastic as it precludes normal diffusion. Specifically, such a resetting can induce a constant asymptotic mean squared displacement in the Markovian case or two distinct regimes of subdiffusive motion in the non-Markovian case.

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.

Fick-Jacobs description and first passage dynamics for diffusion in a channel under stochastic resetting

The transport of particles through channels is of paramount importance in physics, chemistry, and surface science due to its broad real world applications. Much insight can be gained by observing the transition paths of a particle through a channel and collecting statistics on the lifetimes in the channel or the escape probabilities from the channel. In this paper, we consider the diffusive transport through a narrow conical channel of a Brownian particle subject to intermittent dynamics, namely, stochastic resetting. As such, resetting brings the particle back to a desired location from where it resumes its diffusive phase. To this end, we extend the Fick-Jacobs theory of channel-facilitated diffusive transport to resetting-induced transport. Exact expressions for the conditional mean first passage times, escape probabilities, and the total average lifetime in the channel are obtained, and their behavior as a function of the resetting rate is highlighted. It is shown that resetting can expedite the transport through the channel-rigorous constraints for such conditions are then illustrated. Furthermore, we observe that a carefully chosen resetting rate can render the average lifetime of the particle inside the channel minimal. Interestingly, the optimal rate undergoes continuous and discontinuous transitions as some relevant system parameters are varied. The validity of our one-dimensional analysis and the corresponding theoretical predictions is supported by three-dimensional Brownian dynamics simulations. We thus believe that resetting can be useful to facilitate particle transport across biological membranes-a phenomenon that can spearhead further theoretical and experimental studies.

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.

Conditioned backward and forward times of diffusion with stochastic resetting: A renewal theory approach

Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward B and forward F times being the times since the last and until the next reset, respectively, given that the current state of the system X(t) is known. These magnitudes are introduced with the paradigmatic case of diffusion under resetting, for which the backward and forward times are conditioned to the position of the walker. We find analytical expressions for the conditioned backward and forward time probability density functions (PDFs), and we compare them with numerical simulations. The general expressions allow us to study particular scenarios. For instance, for power-law reset time PDFs such that φ(t)∼t^{-1-α}, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of α due to the composition between the long-time scaling of diffusion P(x,t)∼1/sqrt[t] and the reset time PDF.

General approach to stochastic resetting

We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that mean square displacement relaxes to a constant only when the distribution of reset times possess finite mean and variance. In this case, the leading order contribution to the probability density function (PDF) of a Gaussian propagator under resetting exhibits a cusp independent of the specific details of the reset time distribution. For subdiffusion we derive the PDF in Laplace space for arbitrary resetting protocol. Resetting at constant rate allows evaluation of the PDF in terms of H function. We analyze the steady state and derive the rate function governing the relaxation behavior. For a subdiffusive process the steady state could exist even if the distribution of reset times possesses only finite mean.

Gaussian process and Lévy walk under stochastic noninstantaneous resetting and stochastic rest

A stochastic process with movement, return, and rest phases is considered in this paper. For the movement phase, the particles move following the dynamics of the Gaussian process or the ballistic type of Lévy walk, and the time of each movement is random. For the return phase, the particles will move back to the origin with a constant velocity or acceleration or under the action of a harmonic force after each movement, so that this phase can also be treated as a noninstantaneous resetting. After each return, a rest with a random time at the origin follows. The asymptotic behaviors of the mean-squared displacements with different kinds of movement dynamics, returning, and random resting time are discussed. The stationary distributions are also considered when the process is localized. In addition, the mean first passage time is considered when the dynamic of the movement phase is Brownian motion.

One-dimensional telegraphic process with noninstantaneous stochastic resetting

In this paper, we consider the one-dimensional dynamical evolution of a particle traveling at constant speed and performing, at a given rate, random reversals of the velocity direction. The particle is subject to stochastic resetting, meaning that at random times it is forced to return to the starting point. Here we consider a return mechanism governed by a deterministic law of motion, so that the time cost required to return is correlated to the position occupied at the time of the reset. We show that in such conditions the process reaches a stationary state which, for some kinds of deterministic return dynamics, is independent of the return phase. Furthermore, we investigate the first-passage properties of the system and provide explicit formulas for the mean first-hitting time. Our findings are supported by numerical simulations.

Stochastic resetting by a random amplitude

Stochastic resetting, a diffusive process whose amplitude is reset to the origin at random times, is a vividly studied strategy to optimize encounter dynamics, e.g., in chemical reactions. Here we generalize the resetting step by introducing a random resetting amplitude such that the diffusing particle may be only partially reset towards the trajectory origin or even overshoot the origin in a resetting step. We introduce different scenarios for the random-amplitude stochastic resetting process and discuss the resulting dynamics. Direct applications are geophysical layering (stratigraphy) and population dynamics or financial markets, as well as generic search processes.

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.

Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting

We compute exactly the mean perimeter and the mean area of the convex hull of a two-dimensional isotropic Brownian motion of duration t and diffusion constant D, in the presence of resetting to the origin at a constant rate r. We show that for any t, the mean perimeter is given by 〈L(t)〉=2πsqrt[D/r]f_{1}(rt) and the mean area is given by 〈A(t)〉=2πD/rf_{2}(rt) where the scaling functions f_{1}(z) and f_{2}(z) are computed explicitly. For large t≫1/r, the mean perimeter grows extremely slowly as 〈L(t)〉∝ln(rt) with time. Likewise, the mean area also grows slowly as 〈A(t)〉∝ln^{2}(rt) for t≫1/r. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times due to the isotropy of the Brownian motion. Numerical simulations are in perfect agreement with our analytical predictions.

Continuous time random walks under Markovian resetting

We investigate the effects of Markovian resetting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power-law probability density functions. We prove the existence of a nonequilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power-law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset rate, the distance of the initial position to the target, and the characteristic transport exponents.

Target competition for resources under multiple search-and-capture events with stochastic resetting

We develop a general framework for analysing the distribution of resources in a population of targets under multiple independent search-and-capture events. Each event involves a single particle executing a stochastic search that resets to a fixed location x r at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns to x r , where it is reloaded with cargo and a new round of search and capture begins. Using renewal theory, we determine the mean number of resources in each target as a function of the splitting probabilities and unconditional mean first passage times of the corresponding search process without resetting. We then use asymptotic PDE methods to determine the effects of resetting on the distribution of resources generated by diffusive search in a bounded two-dimensional domain with N small interior targets. We show that slow resetting increases the total number of resources M tot across all targets provided that ∑ j = 1 N G ( x r , x j ) < 0 , where G is the Neumann Green's function and x j is the location of the j-th target. This implies that M tot can be optimized by varying r. We also show that the k-th target has a competitive advantage if ∑ j = 1 N G ( x r , x j ) > N G ( x r , x k ) .

Search processes with stochastic resetting and multiple targets

Search processes with stochastic resetting provide a general theoretical framework for understanding a wide range of naturally occurring phenomena. Most current models focus on the first-passage-time problem of finding a single target in a given search domain. Here we use a renewal method to derive general expressions for the splitting probabilities and conditional mean first passage times (MFPTs) in the case of multiple targets. Our analysis also incorporates the effects of delays arising from finite return times and refractory periods. Carrying out a small-r expansion, where r is the mean resetting rate, we obtain general conditions for when resetting increases the splitting probability or reduces the conditional MFPT to a particular target. This also depends on whether π_{tot}=1 or π_{tot}<1, where π_{tot} is the probability that the particle is eventually absorbed by one of the targets in the absence of resetting. We illustrate the theory by considering two distinct examples. The first consists of an actin-rich cell filament (cytoneme) searching along a one-dimensional array of target cells, a problem for which the splitting probabilities and MFPTs can be calculated explicitly. In particular, we highlight how the resetting rate plays an important role in shaping the distribution of splitting probabilities along the array. The second example involves a search process in a three-dimensional bounded domain containing a set of N small interior targets. We use matched asymptotics and Green's functions to determine the behavior of the splitting probabilities and MFPTs in the small-r regime. In particular, we show that the splitting probabilities and MFPTs depend on the "shape capacitance" of the targets.

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.

Queueing theory of search processes with stochastic resetting

We use queueing theory to develop a general framework for analyzing search processes with stochastic resetting, under the additional assumption that following absorption by a target, the particle (searcher) delivers a packet of resources to the target and the search process restarts at the reset point x_{r}. This leads to a sequence of search-and-capture events, whereby resources accumulate in the target under the combined effects of resource supply and degradation. Combining the theory of G/M/∞ queues with a renewal method for analyzing resetting processes, we derive general expressions for the mean and variance of the number of resource packets within the target at steady state. These expressions apply to both exponential and nonexponential resetting protocols and take into account delays arising from various factors such as finite return times, refractory periods, and delays due to the loading or unloading of resources. In the case of exponential resetting, we show how the resource statistics can be expressed in terms of the MFPTs T_{r}(x_{r}) and T_{r+γ}(x_{r}), where r is the resetting rate and γ is the degradation rate. This allows us to derive various general results concerning the dependence of the mean and variance on the parameters r,γ. Our results are illustrated using several specific examples. Finally, we show how fluctuations can be reduced either by allowing the delivery of multiple packets that degrade independently or by having multiple independent searchers.

Brownian motion under noninstantaneous resetting in higher dimensions

We consider Brownian motion under resetting in higher dimensions for the case when the return of the particle to the origin occurs at a constant speed. We investigate the behavior of the probability density function (PDF) and of the mean-squared displacement (MSD) in this process. We study two different resetting protocols: exponentially distributed time intervals between the resetting events (Poissonian resetting) and resetting at fixed time intervals (deterministic resetting). We moreover discuss a general problem of the invariance of the PDF with respect to the return speed, as observed in the one-dimensional system for Poissonian resetting, and show that this one-dimensional situation is the only one in which such an invariance can be found. However, the invariance of the MSD can still be observed in higher dimensions.

Continuous-time random walks under power-law resetting

We study continuous-time random walks (CTRW) with power-law distribution of waiting times under resetting which brings the walker back to the origin, with a power-law distribution of times between the resetting events. Two situations are considered. Under complete resetting, the CTRW after the resetting event starts anew, with a new waiting time, independent of the prehistory. Under incomplete resetting, the resetting of the coordinate does not influence the waiting time until the next jump. We focus on the behavior of the mean-squared displacement (MSD) of the walker from its initial position, on the conditions under which the probability density functions of the walker's displacement show universal behavior, and on this universal behavior itself. We show, that the behavior of the MSD is the same as in the scaled Brownian motion (SBM), being the mean-field model of the CTRW. The intermediate asymptotics of the probability density functions (PDF) for CTRW under complete resetting (provided they exist) are also the same as in the corresponding case for SBM. For incomplete resetting, however, the behavior of the PDF for CTRW and SBM is vastly different.

Resetting processes with noninstantaneous return

We consider a random two-phase process which we call a reset-return one. The particle starts its motion at the origin. The first, displacement, phase corresponds to a stochastic motion of a particle and is finished at a resetting event. The second, return, phase corresponds to the particle's motion toward the origin from the position it attained at the end of the displacement phase. This motion toward the origin takes place according to a given equation of motion. The whole process is a renewal one. We provide general expressions for the stationary probability density function of the particle's position and for the mean hitting time in one dimension. We perform explicit analysis for the Brownian motion during the displacement phase and three different types of the return motion: return at a constant speed, return at a constant acceleration with zero initial speed, and return under the action of a harmonic force. We assume that the waiting times for resetting events follow an exponential distribution or that resetting takes place after a fixed waiting period. For the first two types of return motion and the exponential resetting, the stationary probability density function of the particle's position is invariant under return speed (acceleration), while no such invariance is found for deterministic resetting, and for exponential resetting with return under the action of the harmonic force. We discuss necessary conditions for such invariance of the stationary PDF of the positions with respect to the properties of the return process, and we demonstrate some additional examples when this invariance does or does not take place.