We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results restricted to Markovian random walks, we here find that, depending on the value of the relative standard deviation of the waiting time probability, resetting can be either (i) never beneficial, (ii) beneficial depending on the distance of the reset to the boundary, or (iii) always beneficial.
We investigate the effects of markovian resseting 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 non-equilibrium 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 ra…
We consider Markovian open quantum systems subject to stochastic resetting, which means that the dissipative time evolution is reset at randomly distributed times to the initial state. We show that the ensuing dynamics is non-Markovian and has the form of a generalized Lindblad equation. Interestingly, the statistics of quantum-jumps can be exactly derived. This is achieved by combining techniques from the thermodynamics of quantum-jump trajectories with the renewal structure of the resetting dynamics. We consider as an application of our analysis a driven two-level and an intermittent three-l…
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 and forward times for stochastic processes with resetting to be the times since the last and until the next reset, given that the current state of the system is known. We focus on studying diffusion under Markovian and non-Markovian resetting. For these cases, we find the …
We study the diffusive transport of Markovian random walks on arbitrary networks with stochastic resetting to multiple nodes. We deduce analytical expressions for the stationary occupation probability and for the mean and global first passage times. This general approach allows us to characterize the effect of resetting on the capacity of random walk strategies to reach a particular target or to explore the network. Our formalism holds for ergodic random walks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the efficiency of…
We study the diffusive motion of a test particle in a two-dimensional comb structure consisting of a main backbone channel with continuously distributed side branches, in the presence of stochastic Markovian resetting to the initial position of the particle. We assume that the motion along the infinitely long branches is biased by a confining potential. The crossover to the steady state is quantified in terms of a large deviation function, which is derived for the first time for comb structures in present paper. We show that the relaxation region is demarcated by a nonlinear "light-cone" beyon…
We put forward a novel approach to study the evolution of an arbitrary open quantum system under a resetting process. Using the framework of renewal equations, we find a universal behavior for the mean first return time that goes beyond unitary dynamics and Markovian measurements. Our results show a non-trivial behavior of the mean switching times with the mean measurement time $τ$, which permits tuning $τ$ for minimizing the mean transition time between states. We complement our results with a numerical analysis and we benchmark the results against the corresponding analytical study for low d…
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 $γ_{\rm opt}$ at which the MTA is a minimum, where $r_c=\sqrt {d/\left( {d + 4} \right)} R$ and $R$ is the radius of sphere. As $r$ increases, $γ_{\rm opt}$ exhibits a continuous transition from ze…
We analyze two models of subdiffusion with stochastic resetting. Each of them consists of two parts: subdiffusion based on the continuous-time random walk (CTRW) scheme and independent resetting events generated uniformly in time according to the Poisson point process. In the first model the whole process is reset to the initial state, whereas in the second model only the position is subject to resets. The distinction between these two models arises from the non-Markovian character of the subdiffusive process. We derive exact expressions for the two lowest moments of the full propagator, stati…
We consider the dynamics of lattice random walks with resetting. The walker moving randomly on a lattice of arbitrary dimensions resets at every time step to a given site with a constant probability $r$. We construct a discrete renewal equation and present closed-form expressions for different quantities of the resetting dynamics in terms of the underlying reset-free propagator or Green's function. We apply our formalism to the biased random walk dynamics in one-dimensional unbounded space and show how one recovers in the continuous limits results for diffusion with resetting. The resetting dy…
Hitting the blinking target under stochastic resetting
Jagiellonian University（雅盖隆大学）；Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research, Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University（雅盖隆大学物理、天文与应用科学学院理论与物理研究所及马克·卡克复杂系统研究中心）；Faculty of Computer Science, AGH University of Krakow（克拉科夫AGH科技大学计算机学院）；Academic Computer Center Cyfronet AGH（AGH学术计算中心Cyfronet）；Department of Applied Mathematics, Wrocław University of Science and Technology（弗罗茨瓦夫理工大学应用数学系）；Institute of Physics, Humboldt University of Berlin（柏林洪堡大学物理研究所）
We study the Brownian motion of a particle in a bounded circular 2-dimensional domain, in search for a stationary target on the boundary of the domain. The process switches between two modes: one where it performs a two-dimensional diffusion inside the circle and one where it travels along the one-dimensional boundary. During the diffusion, the Brownian particle resets to its initial position with a constant rate $r$. The Fokker-Planck formalism allows us to calculate the mean time to absorption (MTA) as well as the optimal resetting rate for which the MTA is minimized. From the derived analyt…
What determines the average length of a queue which stretches in front of a service station? The answer to this question clearly depends on the average rate at which jobs arrive at the queue and on the average rate of service. Somewhat less obvious is the fact that stochastic fluctuations in service and arrival times are also important, and that these are a major source of backlogs and delays. Strategies that could mitigate fluctuations induced delays are in high demand as queue structures appear in various natural and man-made systems. Here we demonstrate that a simple service resetting mecha…
In this paper, we investigate the effects of stochastic resetting on diffusion in $\R^d\backslash \calU$, where $\calU$ is a bounded obstacle with a partially absorbing surface $\partial \calU$. We begin by considering a Robin boundary condition with a constant reactivity $κ_0$, and show how previous results are recovered in the limits $κ_0\rightarrow 0,\infty$. We then generalize the Robin boundary condition to a more general probabilistic model of diffusion-mediated surface reactions using an encounter-based approach. The latter considers the joint probability density or propagator $P(\x,\el…
We consider a particle undergoing diffusion with stochastic resetting in a bounded domain $\calU\subset \R^d$ for $d=2,3$. The domain is perforated by a set of partially absorbing targets within which the particle may be absorbed at a rate $κ$. Each target is assumed to be much smaller than $|\calU|$, which allows us to use asymptotic and Green's function methods to solve the diffusion equation in Laplace space. In particular, we construct an inner solution within the interior and local exterior of each target, and match it with an outer solution in the bulk of $\calU$. This yields an asymptot…
We investigate the Brownian diffusion of particles in one spatial dimension and in the presence of finite regions within which particles can either evaporate or be reset to a given location. For open boundary conditions, we highlight the appearance of a Brownian yet non-Gaussian diffusion: at long times, the particle distribution is non-Gaussian but its variance grows linearly in time. Moreover, we show that the effective diffusion coefficient of the particles in such systems is bounded from below by $(1-2/π)$ times their bare diffusion coefficient. For periodic boundary conditions, i.e., for …
What happens when the time evolution of a fluctuating interface is interrupted with resetting to a given initial configuration after random time intervals $τ$ distributed as a power-law $\sim τ^{-(1+α)};~α> 0$? For an interface of length $L$ in one dimension, and an initial flat configuration, we show that depending on $α$, the dynamics as $L \to \infty$ exhibits a rich long-time behavior. Without resetting, the interface width grows unbounded with time as $t^β$, where $β$ is the so-called growth exponent. We show that introducing resetting induces for $α>1$ and at long times fluctuations that…
In this paper we consider the diffusive search for a bounded target $Ω\in \R^d$ with its boundary $\partial Ω$ totally absorbing. We assume that the target is surrounded by a semipermeable interface given by the closed surface $\partial \calM$ with $Ω\subset \calM\subset \R^d$. That is, the interface totally surrounds the target and thus partially screens the diffusive search process. We also assume that the position of the diffusing particle (searcher) randomly resets to its initial position $\x_0$ according to a Poisson process with a resetting rate $r$. The location $\x_0$ is taken to be ou…
A powerful strategy to accelerate quantum-walk-based search algorithms leverages on resetting protocols, where a detector monitors a target site and the evolution of the walker is restarted if no detection occurs within a fixed time interval. The optimal resetting rate can be extracted from the time evolution of the probability $S(t)$ that the detector has not clicked up to time $t$. We analyze $S(t)$ for a quantum walk on a one-dimensional lattice when the coupling between sites decays algebraically as $d^{-α}$ with the distance $d$, for $α\in(0,\infty)$. At long times, $S(t)$ decays with a u…
In this paper we use asymptotic analysis to determine the steady-state mean number of resources in each of $N$ small interior targets within a three-dimensional bounded domain. The accumulation of resources is based on multiple rounds of search-and-capture events; whenever a searcher finds a target it delivers a resource packet to the target, after which it escapes and returns to its initial position (resetting after capture). The searcher is then resupplied with cargo and a new search process is initiated after a random delay. Assuming that the accumulation of resources is counterbalanced by …
We solve an adaptive search model where a random walker or Lévy flight stochastically resets to previously visited sites on a $d$-dimensional lattice containing one trapping site. Due to reinforcement, a phase transition occurs when the resetting rate crosses a threshold above which non-diffusive stationary states emerge, localized around the inhomogeneity. The threshold depends on the trapping strength and on the walker's return probability in the memoryless case. The transition belongs to the same class as the self-consistent theory of Anderson localization. These results show that similarly…
Generalized empirical currents represent a vast class of thermodynamic observables of mesoscopic systems. Their fluctuations satisfy the thermodynamic uncertainty relations (TURs), as they can be bounded by the average entropy production. Here, we derive a general closed expression for the hyperaccurate current in discrete-state Markovian systems, i.e., the one with the least fluctuations, for both discrete- and continuous-time evolution. We show that its associated hyperaccurate bound is generally much tighter than the one given by the TURs, and might be crucial to providing a reliable estima…
The aim of the paper is to understand how the inclusion of more and more time-scales into a stochastic stationary Markovian process affects its conditional probability. To this end, we consider two Gaussian processes: (i) a short-range correlated process with an infinite set of time-scales bounded from below, and (ii) a power-law correlated process with an infinite and unbounded set of time-scales. For these processes we investigate the equal position conditional probability P(x,t|x,0) and the mean First Passage Time T(L). The function P(x,t|x,0) can be considered as a proxy of the persistence…
The aim of this paper is to investigate how the correlation properties of a stationary Markovian stochastic processes affect the First Passage Time distribution. First Passage Time issues are a classical topic in stochastic processes research. They also have relevant applications, for example, in many fields of finance such as the assessment of the default risk for firms' assets. By using some explicit examples, in this paper we will show that the tail of the First Passage Time distribution crucially depends on the correlation properties of the process and it is independent from its stationary…
The thermodynamic uncertainty relation, which establishes a universal trade-off between nonequilibrium current fluctuations and dissipation, has been found for various Markovian systems. However, this relation has not been revealed for non-Markovian systems; therefore, we investigate the thermodynamic uncertainty relation for time-delayed Langevin systems. We prove that the fluctuation of arbitrary dynamical observables is constrained by the Kullback--Leibler divergence between the distributions of the forward path and its reversed counterpart. Specifically, for observables that are antisymmet…
