Correlated continuous-time random walk with stochastic resetting

Abstract

It is known that the introduction of stochastic resetting in an uncorrelated random walk process can lead to the emergence of a stationary state, i.e. the diffusion evolves towards a saturation state, and a steady Laplace distribution is reached. In this paper, we turn to study the anomalous diffusion of the correlated continuous-time random walk considering stochastic resetting. Results reveal that it displays quite different diffusive behaviors from the uncorrelated one. For the weak correlation case, the stochastic resetting mechanism can slow down the diffusion. However, for the strong correlation case, we find that the stochastic resetting cannot compete with the space-time correlation, and the diffusion presents the same behaviors with the one without resetting. Meanwhile, a steady distribution is never reached.

The Lévy walk with rests under stochastic resetting

“…Finally, we note that for the special case 0 < α < 1 < β < 2, considering that the state occupation mechanism makes the LWR present subdiffusive behavior, it is not difficult to conclude that just like the subdiffusive CTRW under resetting [52,60,61], the diffusion is hindered by the introduction of the stochastic resetting and finally evolves toward a saturation state. By considering various pairs of α and β, the numerical results of the MSD ⟨∆x 2 (t)⟩ varying with time t are displayed in figure 3 and it can be seen that the numerical results are all well consistent with the theoretical conclusions.…”

“…Therefore, for 1 < min{α, β} < 2, the ∼t 2−β+min{α,β} term will dominate the diffusion for large time limit; in other words, the introduction of the stochastic resetting surprisingly enhances the LWR's diffusion. As far as we know, after considering resetting, the diffusion of the stochastic process may be weakened, such as in the subdiffusive continuous time random walk (CTRW) [52,60,61], or remain unchanged, such as in the LW [60,61]. The enhancement effect caused by the stochastic resetting has not yet been reported.…”

“…Meanwhile, transport and diffusion of classical stochastic processes and models under stochastic resetting have been intensively studied, e.g. continuous-time random walks [49][50][51][52][53], Lévy flights [54,55], fractional Brownian motion [56,57], scaled Brownian motion [58,59] and LW [60][61][62]. Quantum systems under stochastic resetting are studied as well [63][64][65] and, recently, the diffusion processes under stochastic resetting has been realized experimentally [66,67].…”

“…As far as we know, as a theory to describe the intermittent search strategy, the LWR under stochastic resetting has not been discussed yet. In fact, considering that other classical stochastic processes and models under stochastic resetting have been fully studied [49][50][51][52][53][54][55][56][57][58][59][60][61][62], it is quite surprising, as a widely applied theoretical model in physics and biology, the LWR under stochastic resetting is still unexplored. In this manuscript, we aim to fill this gap.…”

Fractional advection diffusion asymmetry equation, derivation, solution and application

“…When the sojourn time PDF has a fat tail and the variance of the displacement is finite, a wide range of anomalous behaviors emerge [3,63]. As mentioned, here we will focus on the widely applicable case 1 < α < 2 [64][65][66][67][68][69][70][71].…”