Robust random search with scale-free stochastic resetting

Extremal statistics for a resetting Brownian motion before its first-passage time

We study the extreme value statistics of a one-dimensional resetting Brownian motion (RBM) till its first passage through the origin starting from the position x_{0} (>0). By deriving the exit probability of RBM in an interval [0,M] from the origin, we obtain the distribution P_{r}(M|x_{0}) of the maximum displacement M and thus gives the expected value 〈M〉 of M as functions of the resetting rate r and x_{0}. We find that 〈M〉 decreases monotonically as r increases, and tends to 2x_{0} as r→∞. In the opposite limit, 〈M〉 diverges logarithmically as r→0. Moreover, we derive the propagator of RBM in the Laplace domain in the presence of both absorbing ends, and then leads to the joint distribution P_{r}(M,t_{m}|x_{0}) of M and the time t_{m} at which this maximum is achieved in the Laplace domain by using a path decomposition technique, from which the expected value 〈t_{m}〉 of t_{m} is obtained explicitly. Interestingly, 〈t_{m}〉 shows a nonmonotonic dependence on r, and attains its minimum at an optimal r^{*}≈2.71691D/x_{0}^{2}, where D is the diffusion coefficient. Finally, we perform extensive simulations to validate our theoretical results.

Universal performance bounds of restart

As has long been known to computer scientists, the performance of probabilistic algorithms characterized by relatively large runtime fluctuations can be improved by applying a restart, i.e., episodic interruption of a randomized computational procedure followed by initialization of its new statistically independent realization. A similar effect of restart-induced process acceleration could potentially be possible in the context of enzymatic reactions, where dissociation of the enzyme-substrate intermediate corresponds to restarting the catalytic step of the reaction. To date, a significant number of analytical results have been obtained in physics and computer science regarding the effect of restart on the completion time statistics in various model problems, however, the fundamental limits of restart efficiency remain unknown. Here we derive a range of universal statistical inequalities that offer constraints on the effect that restart could impose on the completion time of a generic stochastic process. The corresponding bounds are expressed via simple statistical metrics of the original process such as harmonic mean h, median value m, and mode M, and, thus, are remarkably practical. We test our analytical predictions with multiple numerical examples, discuss implications arising from them and important avenues of future work.

First-passage time of run-and-tumble particles with noninstantaneous resetting

We study the statistics of the first-passage time of a single run-and-tumble particle (RTP) in one spatial dimension, with or without resetting, to a fixed target located at L>0. First, we compute the first-passage time distribution of a free RTP, without resetting or in a confining potential, but averaged over the initial position drawn from an arbitrary distribution p(x). Recent experiments used a noninstantaneous resetting protocol that motivated us to study in particular the case where p(x) corresponds to the stationary non-Boltzmann distribution of an RTP in the presence of a harmonic trap. This distribution p(x) is characterized by a parameter ν>0, which depends on the microscopic parameters of the RTP dynamics. We show that the first-passage time distribution of the free RTP, drawn from this initial distribution, develops interesting singular behaviors, depending on the value of ν. We then switch on resetting, mimicked by relaxation of the RTP in the presence of a harmonic trap. Resetting leads to a finite mean first-passage time and we study this as a function of the resetting rate for different values of the parameters ν and b=L/c, where c is the position of the right edge of the initial distribution p(x). In the diffusive limit of the RTP dynamics, we find a rich phase diagram in the (b,ν) plane, with an interesting reentrance phase transition. Away from the diffusive limit, qualitatively similar rich behaviors emerge for the full RTP dynamics.

Random walks on complex networks under time-dependent stochastic resetting

We study discrete-time random walks on networks subject to a time-dependent stochastic resetting, where the walker either hops randomly between neighboring nodes with a probability 1-ϕ(a) or is reset to a given node with a complementary probability ϕ(a). The resetting probability ϕ(a) depends on the time a since the last reset event (also called a, the age of the walker). Using the renewal approach and spectral decomposition of the transition matrix, we formulate the stationary occupation probability of the walker at each node and the mean first passage time between two arbitrary nodes. Concretely, we consider two different time-dependent resetting protocols that are both exactly solvable. One is that ϕ(a) is a step-shaped function of a and the other one is that ϕ(a) is a rational function of a. We demonstrate the theoretical results on several different networks, also validated by numerical simulations, and find that the time-modulated resetting protocols can be more advantageous than the constant-probability resetting in accelerating the completion of a target search process.

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.

Fat tails and black swans: Exact results for multiplicative processes with resets

We consider a class of multiplicative processes which, added with stochastic reset events, give origin to stationary distributions with power-law tails-ubiquitous in the statistics of social, economic, and ecological systems. Our main goal is to provide a series of exact results on the dynamics and asymptotic behavior of increasingly complex versions of a basic multiplicative process with resets, including discrete and continuous-time variants and several degrees of randomness in the parameters that control the process. In particular, we show how the power-law distributions are built up as time elapses, how their moments behave with time, and how their stationary profiles become quantitatively determined by those parameters. Our discussion emphasizes the connection with financial systems, but these stochastic processes are also expected to be fruitful in modeling a wide variety of social and biological phenomena.

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

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