Diffusion with stochastic resetting at power-law times

Abstract

What happens when a continuously evolving stochastic process is interrupted with large changes at random intervals τ distributed as a power law ∼τ^{-(1+α)};α>0? Modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition, we obtain exact closed-form expressions for both static and dynamic quantities, while accounting for strong correlations implied by a power law. Our results show that the resulting dynamics exhibits a spectrum of rich long-time behavior, from an ever-spreading spatial distribution for α<1, to one that is time independent for α>1. The dynamics has strong consequences on the time to reach a distant target for the first time; we specifically show that there exists an optimal α that minimizes the mean time to reach the target, thereby offering a step towards a viable strategy to locate targets in a crowded environment.

Symmetric exclusion process under stochastic resetting

“…The variance of the diffusive current σ 2 d (t) = J 2 d (t) − J d (t) 2 can be obtained using Eqs. (39) and (41). In particular, in the long time limit, the variance increases linearly with time t, and is given by, Distribution of J d : It is interesting to investigate the probability distribution of the diffusive current J d (t).…”

Transport properties of random walks under stochastic noninstantaneous resetting

“…In this section we consider a Gaussian propagator and a Pareto for the resetting time PDF. For practical examples of phenomena that may generate scale-free reset times we refer the reader to [14]. The interest of this distribution is that the exponent γ controls the existence of finite moments.…”

“…There, a diffusive particle is studied when it may occasionally reset its position with a constant probability and the authors find that a non-equilibrium steady state (NESS) is reached and the mean first passage time of the overall process is finite and attains a minimum in terms of the resetting rate. The existence of a NESS has been further studied for different types of motion and resetting mechanisms [5][6][7][8][9][10][11][12][13][14][15][16][17][18], showing that they are not exclusive of diffusion with Markovian resets. Aside from these, other works have shown that the resetting does not always generate a NESS but transport is also possible when the resetting probability density function (PDF) is long-tailed [19][20][21][22] or when the resetting process is subordinated to the motion [10,12].…”

Anomalous Diffusion in Random-Walks With Memory-Induced Relocations

“…Thereafter, this property has been confirmed by numerous works on Markovian resets in different contexts, as multi-dimensional diffusion [11], coagulation-diffusion processes [12], confined diffusion [13,14], diffusion with a refractory period after the resets [15], anomalous subdiffusion [16,17], monotonic stochastic motion [18,19], continuous-time random walk (CTRW) velocity models [20], the telegraphic process [21] and underdamped Brownian motion [22]. Likewise, in [23], a steady state is shown to appear when a diffusion process is restarted at a time-dependent rate and in [24] power-law reset time probability density functions (pdf) are considered and conditions for a steady state to exist are found. Finally, general conditions on the reset time pdf for the appearance of a steady state have been found in [25,26].…”