Localization Transition Induced by Learning in Random Searches

Quest for optimal quantum resetting: Protocols for a particle on a chain

In the classical context, it is well known that, sometimes, if a search does not find its target, it is better to start the process anew. This is known as resetting. The quantum counterpart of resetting also indicates speeding up the detection process by eliminating the dark states, i.e., situations in which the particle avoids detection. In this work, we introduce the most probable position resetting (MPR) protocol, in which, at a given resetting step, resets are done with certain probabilities to the set of possible peak positions (where the probability of finding the particle is maximum) that could occur because of the previous resets and followed by uninterrupted unitary evolution, irrespective of which path was taken by the particle in previous steps. In a tight-binding lattice model, there exists a twofold degeneracy (left and right) of the positions of maximum probability. The survival probability with optimal restart rate approaches 0 (detection probability approaches 1) when the particle is reset with equal probability on both sides path independently. This protocol significantly reduces the optimal mean first-detected-passage time (FDT), and it performs better even if the detector is far apart compared to the usual resetting protocols in which the particle is brought back to the initial position. We propose a modified protocol, an adaptive two-stage MPR, by making the associated probabilities of going to the right and left a function of steps. In this protocol, we see a further reduction of the optimal mean FDT and improvement in the search process when the detector is far apart.

Instability in the quantum restart problem

Repeatedly monitored quantum walks with a rate 1/τ yield discrete-time trajectories which are inherently random. With these paths the first-hitting time with sharp restart is studied. We find an instability in the optimal mean hitting time, which is not found in the corresponding classical random-walk process. This instability implies that a small change in parameters can lead to a rather large change of the optimal restart time. We show that the optimal restart time versus τ, as a control parameter, exhibits sets of staircases and plunges. The plunges, are due to the mentioned instability, which in turn is related to the quantum oscillations of the first-hitting time probability, in the absence of restarts. Furthermore, we prove that there are only two patterns of staircase structures, dependent on the parity of the distance between the target and the source in units of lattice constant. The global minimum of the hitting time is controlled not only by the restart time, as in classical problems, but also by the sampling time τ. We provide numerical evidence that this global minimum occurs for the τ minimizing the mean hitting time, given restarts taking place after each measurement. Last, we numerically show that the instability found in this work is relatively robust against stochastic perturbations in the sampling time τ.

Active particle in one dimension subjected to resetting with memory

The study of diffusion with preferential returns to places visited in the past has attracted increased attention in recent years. In these highly non-Markov processes, a standard diffusive particle intermittently resets at a given rate to previously visited positions. At each reset, a position to be revisited is randomly chosen with a probability proportional to the accumulated amount of time spent by the particle at that position. These preferential revisits typically generate a very slow diffusion, logarithmic in time, but still with a Gaussian position distribution at late times. Here we consider an active version of this model, where between resets the particle is self-propelled with constant speed and switches direction in one dimension according to a telegraphic noise. Hence there are two sources of non-Markovianity in the problem. We exactly derive the position distribution in Fourier space, as well as the variance of the position at all times. The crossover from the short-time ballistic regime, dominated by activity, to the long-time anomalous logarithmic growth induced by memory is studied. We also analytically derive a large deviation principle for the position, which exhibits a logarithmic time scaling instead of the usual algebraic form. Interestingly, at large distances, the large deviations become independent of time and match the nonequilibrium steady state of a particle under resetting to its starting position only.

Universal and nonuniversal probability laws in Markovian open quantum dynamics subject to generalized reset processes

We consider quantum-jump trajectories of Markovian open quantum systems subject to stochastic in time resets of their state to an initial configuration. The reset events provide a partitioning of quantum trajectories into consecutive time intervals, defining sequences of random variables from the values of a trajectory observable within each of the intervals. For observables related to functions of the quantum state, we show that the probability of certain orderings in the sequences obeys a universal law. This law does not depend on the chosen observable and, in the case of Poissonian reset processes, not even on the details of the dynamics. When considering (discrete) observables associated with the counting of quantum jumps, the probabilities in general lose their universal character. Universality is only recovered in cases when the probability of observing equal outcomes in the same sequence is vanishingly small, which we can achieve in a weak-reset-rate limit. Our results extend previous findings on classical stochastic processes [N. R. Smith et al., Europhys. Lett. 142, 51002 (2023)0295-507510.1209/0295-5075/acd79e] to the quantum domain and to state-dependent reset processes, shedding light on relevant aspects for the emergence of universal probability laws.

Optimizing the random search of a finite-lived target by a Lévy flight

In many random search processes of interest in chemistry, biology, or during rescue operations, an entity must find a specific target site before the latter becomes inactive, no longer available for reaction or lost. We present exact results on a minimal model system, a one-dimensional searcher performing a discrete time random walk, or Lévy flight. In contrast with the case of a permanent target, the capture probability and the conditional mean first passage time can be optimized. The optimal Lévy index takes a nontrivial value, even in the long lifetime limit, and exhibits an abrupt transition as the initial distance to the target is varied. Depending on the target lifetime, this transition is discontinuous or continuous, separated by a nonconventional tricritical point. These results pave the way to the optimization of search processes under time constraints.

Lévy movements and a slowly decaying memory allow efficient collective learning in groups of interacting foragers

Many animal species benefit from spatial learning to adapt their foraging movements to the distribution of resources. Learning involves the collection, storage and retrieval of information, and depends on both the random search strategies employed and the memory capacities of the individual. For animals living in social groups, spatial learning can be further enhanced by information transfer among group members. However, how individual behavior affects the emergence of collective states of learning is still poorly understood. Here, with the help of a spatially explicit agent-based model where individuals transfer information to their peers, we analyze the effects on the use of resources of varying memory capacities in combination with different exploration strategies, such as ordinary random walks and Lévy flights. We find that individual Lévy displacements associated with a slow memory decay lead to a very rapid collective response, a high group cohesion and to an optimal exploitation of the best resource patches in static but complex environments, even when the interaction rate among individuals is low.

Restart Expedites Quantum Walk Hitting Times

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 a 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 sudden drops as the rate of sampling is modified. In the absence of restart and in the Zeno limit, the detection of the walker is not possible, and we examine how restart overcomes this well-known problem, showing that the optimal restart time becomes insensitive to the sampling period.

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.

Unexpected advantages of exploitation for target searches in complex networks

Exploitation universally emerges in various decision-making contexts, e.g., animals foraging, web surfing, the evolution of scientists' research topics, and our daily lives. Despite its ubiquity, exploitation, which refers to the behavior of revisiting previous experiences, has often been considered to delay the search process of finding a target. In this paper, we investigate how exploitation affects search performance by applying a non-Markovian random walk model, where a walker randomly revisits a previously visited node using long-term memory. We analytically study two broad forms of network structures, namely, (i) clique-like networks and (ii) lollipop-like networks and find that exploitation can significantly improve search performance in lollipop-like networks, whereas it hinders target search in clique-like networks. Moreover, we numerically verify that exploitation can reduce the time needed to fully explore the underlying networks using 550 diverse real-world networks. Based on the analytic result, we define the lollipop-likeness of a network and observe a positive relationship between the advantage of exploitation and lollipop-likeness.

Synchronization in the Kuramoto model in presence of stochastic resetting

What happens when the paradigmatic Kuramoto model involving interacting oscillators of distributed natural frequencies and showing spontaneous collective synchronization in the stationary state is subject to random and repeated interruptions of its dynamics with a reset to the initial condition? While resetting to a synchronized state, it may happen between two successive resets that the system desynchronizes, which depends on the duration of the random time interval between the two resets. Here, we unveil how such a protocol of stochastic resetting dramatically modifies the phase diagram of the bare model, allowing, in particular, for the emergence of a synchronized phase even in parameter regimes for which the bare model does not support such a phase. Our results are based on an exact analysis invoking the celebrated Ott-Antonsen ansatz for the case of the Lorentzian distribution of natural frequencies and numerical results for Gaussian frequency distribution. Our work provides a simple protocol to induce global synchrony in the system through stochastic resetting.

Phase Transition in a Non-Markovian Animal Exploration Model with Preferential Returns

We study a non-Markovian and nonstationary model of animal mobility incorporating both exploration and memory in the form of preferential returns. Exact results for the probability of visiting a given number of sites are derived and a practical WKB approximation to treat the nonstationary problem is developed. A mean-field version of this model, first suggested by Song et al., [Modelling the scaling properties of human mobility, Nat. Phys. 6, 818 (2010)NPAHAX1745-247310.1038/nphys1760] was shown to well describe human movement data. We show that our generalized model adequately describes empirical movement data of Egyptian fruit bats (Rousettus aegyptiacus) when accounting for interindividual variation in the population. We also study the probability of visiting any site a given number of times and derive a mean-field equation. Our analysis yields a remarkable phase transition occurring at preferential returns which scale linearly with past visits. Following empirical evidence, we suggest that this phase transition reflects a trade-off between extensive and intensive foraging modes.

Joint statistics of space and time exploration of one-dimensional random walks

The statistics of first-passage times of random walks to target sites has proved to play a key role in determining the kinetics of space exploration in various contexts. In parallel, the number of distinct sites visited by a random walker and related observables has been introduced to characterize the geometry of space exploration. Here, we address the question of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies the coupling between the kinetics and geometry of search trajectories. Focusing on one-dimensional systems, we present a general method and derive explicit expressions of this joint distribution for several representative examples of Markovian search processes. In addition, we obtain a general scaling form, which holds also for non-Markovian processes and captures the general dependence of the joint distribution on its space and time variables. We argue that the joint distribution has important applications to various problems, such as a conditional form of the Rosenstock trapping model, and the persistence properties of self-interacting random walks.

Tuning of the Dielectric Relaxation and Complex Susceptibility in a System of Polar Molecules: A Generalised Model Based on Rotational Diffusion with Resetting

The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.

Run with the Brownian Hare, Hunt with the Deterministic Hounds

We present analytic results for mean capture time and energy expended by a pack of deterministic hounds actively chasing a randomly diffusing prey. Depending on the number of chasers, the mean capture time as a function of the prey's diffusion coefficient can be monotonically increasing, decreasing, or attain a minimum at a finite value. Optimal speed and number of chasing hounds exist and depend on each chaser's baseline power consumption. The model can serve as an analytically tractable basis for further studies with bearing on the growing field of smart microswimmers and autonomous robots.

Hierarchical, Memory-Based Movement Models for Translocated Elk (Cervus canadensis)

The use of spatial memory is well-documented in many animal species and has been shown to be critical for the emergence of spatial learning. Adaptive behaviors based on learning can emerge thanks to an interdependence between the acquisition of information over time and movement decisions. The study of how spatio-ecological knowledge is constructed throughout the life of an individual has not been carried out in a quantitative and comprehensive way, hindered by the lack of knowledge of the information an animal already has of its environment at the time monitoring begins. Identifying how animals use memory to make beneficial decisions is fundamental to developing a general theory of animal movement and space use. Here we propose several mobility models based on memory and perform hierarchical Bayesian inference on 11-month trajectories of 21 elk after they were released in a completely new environment. Almost all the observed animals exhibited preferential returns to previously visited patches, such that memory and random exploration phases occurred. Memory decay was mild or negligible over the study period. The fact that individual elk rapidly become used to a relatively small number of patches was consistent with the hypothesis that they seek places with predictable resources and reduced mortality risks such as predation.

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.

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.

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.

Role of dimensions in first passage of a diffusing particle under stochastic resetting and attractive bias

Recent studies in one dimension have revealed that the temporal advantage rendered by stochastic resetting to diffusing particles in attaining first passage may be annulled by a sufficiently strong attractive potential. We extend the results to higher dimensions. For a diffusing particle in an attractive potential V(R)=kR^{n}, in general d dimensions, we study the critical strength k=k_{c} above which resetting becomes disadvantageous. The point of continuous transition may be exactly found even in cases where the problem with resetting is not solvable, provided the first two moments of the problem without resetting are known. We find the dimensionless critical strength κ_{c,n}(k_{c}) exactly when d/n and 2/n take positive integral values. Also for the limiting case of a box potential (representing n→∞), and the special case of a logarithmic potential kln(R/a), we find the corresponding transition points κ_{c,∞} and κ_{c,l} exactly for any dimension d. The asymptotic forms of the critical strengths at large dimensions d are interesting. We show that for the power law potential, for any n∈(0,∞), the dimensionless critical strength κ_{c,n}∼d^{1/n} at large d. For the box potential, asymptotically, κ_{c,∞}∼(1-ln(d/2)/d), while for the logarithmic potential, κ_{c,l}∼d.

Random walks on networks with stochastic resetting

We study random walks with stochastic resetting to the initial position on arbitrary networks. We obtain the stationary probability distribution as well as the mean and global first passage times, which allow us to characterize the effect of resetting on the capacity of a random walker to reach a particular target or to explore a finite network. We apply the results to rings, Cayley trees, and random and complex networks. Our formalism holds for undirected networks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the search efficiency in different structures with the small-world property or communities. In this way, we extend the study of resetting processes to the domain of networks.

Collective learning from individual experiences and information transfer during group foraging

Living in groups brings benefits to many animals, such as protection against predators and an improved capacity for sensing and making decisions while searching for resources in uncertain environments. A body of studies has shown how collective behaviours within animal groups on the move can be useful for pooling information about the current state of the environment. The effects of interactions on collective motion have been mostly studied in models of agents with no memory. Thus, whether coordinated behaviours can emerge from individuals with memory and different foraging experiences is still poorly understood. By means of an agent-based model, we quantify how individual memory and information fluxes can contribute to improving the foraging success of a group in complex environments. In this context, we define collective learning as a coordinated change of behaviour within a group resulting from individual experiences and information transfer. We show that an initially scattered population of foragers visiting dispersed resources can gradually achieve cohesion and become selectively localized in space around the most salient resource sites. Coordination is lost when memory or information transfer among individuals is suppressed. The present modelling framework provides predictions for empirical studies of collective learning and could also find applications in swarm robotics and motivate new search algorithms based on reinforcement.

Understanding the ontogeny of foraging behaviour: insights from combining marine predator bio-logging with satellite-derived oceanography in hidden Markov models

The development of foraging strategies that enable juveniles to efficiently identify and exploit predictable habitat features is critical for survival and long-term fitness. In the marine environment, meso- and sub-mesoscale features such as oceanographic fronts offer a visible cue to enhanced foraging conditions, but how individuals learn to identify these features is a mystery. In this study, we investigate age-related differences in the fine-scale foraging behaviour of adult (aged ≥ 5 years) and immature (aged 2–4 years) northern gannets Morus bassanus. Using high-resolution GPS-loggers, we reveal that adults have a much narrower foraging distribution than immature birds and much higher individual foraging site fidelity. By conditioning the transition probabilities of a hidden Markov model on satellite-derived measures of frontal activity, we then demonstrate that adults show a stronger response to frontal activity than immature birds, and are more likely to commence foraging behaviour as frontal intensity increases. Together, these results indicate that adult gannets are more proficient foragers than immatures, supporting the hypothesis that foraging specializations are learned during individual exploratory behaviour in early life. Such memory-based individual foraging strategies may also explain the extended period of immaturity observed in gannets and many other long-lived species.

Random walks on intersecting geometries

We present an analytical approach to study simple symmetric random walks on a crossing geometry consisting of a plane square lattice crossed by n_{l} number of lines that all meet each other at a single point (the origin) on the plane. The probability density to find the walker at a given distance from the origin either in a line or in the plane geometry is exactly calculated as a function of time t. We find that the large-time asymptotic behavior of the walker for any arbitrary number n_{l} of lines is eventually governed by the diffusion of the walker on the plane after a crossover time approximately given by t_{c}∝n_{l}^{2}. We show that this competition can be changed in favor of the line geometry by switching on an arbitrarily small perturbation of a drift term in which even a weak biased walk is able to drain the whole probability density into the line at long-time limit. We also present the results of our extensive simulations of the model which perfectly support our analytical predictions. Our method can, however, be simply extended to other crossing geometries with a single common point.

Recurrence time correlations in random walks with preferential relocation to visited places

Random walks with memory usually involve rules where a preference for either revisiting or avoiding those sites visited in the past are introduced somehow. Such effects have a direct consequence on the transport properties as well as on the statistics of first-passage and subsequent recurrence times through a site. A preference for revisiting sites is thus expected to result in a positive correlation between consecutive recurrence times. Here we derive a continuous-time generalization of the random walk model with preferential relocation to visited sites proposed in Phys. Rev. Lett. 112, 240601 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.240601 to explore this effect, together with the main transport properties induced by the long-range memory. Despite the long-range memory effects governing the process, our analytical treatment allows us to (i) observe the existence of an asymptotic logarithmic (ultraslow) growth for the mean square displacement, in accordance to the results found for the original discrete-time model, and (ii) confirm the existence of positive correlations between first-passage and subsequent recurrence times. This analysis is completed with a comprehensive numerical study which reveals, among other results, that these correlations between first-passage and recurrence times also exhibit clear signatures of this ultraslow relaxation dynamics.

Subdiffusive continuous-time random walks with stochastic resetting

We analyze two models of subdiffusion with stochastic resetting. Each of them consists of two parts: subdiffusion based on the continuous-time random walk 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, stationary distributions, and first hitting time statistics. We also show, with an example of a constant drift, how these models can be generalized to include external forces. Possible applications to data analysis and modeling of biological systems are also discussed.

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 under Restart with Branching

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 determine how restart with branching affects the mean completion time of an arbitrary stochastic process. The universality of this result is demonstrated and its connection to extreme value theory is also pointed out and explored.

Random Search with Resetting: A Unified Renewal Approach

We provide a unified renewal approach to the problem of random search for several targets under resetting. This framework does not rely on specific properties of the search process and resetting procedure, allows for simpler derivation of known results, and leads to new ones. Concentrating on minimizing the mean hitting time, we show that resetting at a constant pace is the best possible option if resetting helps at all, and derive the equation for the optimal resetting pace. No resetting may be a better strategy if without resetting the probability of not finding a target decays with time to zero exponentially or faster. We also calculate splitting probabilities between the targets, and define the limits in which these can be manipulated by changing the resetting procedure. We moreover show that the number of moments of the hitting time distribution under resetting is not less than the sum of the numbers of moments of the resetting time distribution and the hitting time distribution without resetting.