Continuous-time random walks and Lévy walks with stochastic resetting

Abstract

Intermittent stochastic processes appear in a wide field, such as chemistry, biology, ecology, and computer science. This paper builds up the theory of intermittent continuous time random walk (CTRW) and Lévy walk, in which the particles are stochastically reset to a given position with a resetting rate r. The mean squared displacements of the CTRW and Lévy walks with stochastic resetting are calculated, uncovering that the stochastic resetting always makes the CTRW process localized and Lévy walk diffuse slower. The asymptotic behaviors of the probability density function of Lévy walk with stochastic resetting are carefully analyzed under different scales of x, and a striking influence of stochastic resetting is observed.

Time-averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

“…Straightforward future developments of this study is to apply the TAMSD-and EB-based formulation (•) to other stochastic processes with a resetting dynamics imposed-such as CTRWs 16, 37,38,55,62,71,124,216 , Lévy walks/flights 217 , SBM 53,54,125 , exponential SBM 131 , diffusion with multiple mobility states, the OU process 214,218 , geometric BM [219][220][221][222][223] , diffusion models with distributed 224,225 and "diffusing diffusivity" (DD) 144,[226][227][228][229][230][231] as well as various "hybrid" processes (SBM-HDPs 126 , FBM-DD 144 , SBM-DD 129 , etc. ), (•) to employ other types of resetting protocols/setups (periodic, power-law, and other functional forms for ψ(r) distributions; resetting when particular x max values are reached, resetting to distributed resetting points, with memory effects, etc.…”

Gaussian Process and Levy Walk under Stochastic Non-instantaneous Resetting and Stochastic Rest

“…Specifically, the particle may reset to a given position x r with a resetting rate r ∈ [0, 1] after each step of moving. The results in [35] indicate that the resetting can make the particles localized.…”

“…8. The asymptotic behavior of MSD in (35) indicates the competition between movement phase and rest phase. In summary, there is always a competition between the movement phase and rest phase.…”

“…For the related discussions, see [32][33][34]. Besides, CTRW and Lévy walks with stochastic resetting have been studied in [7,35]. Specifically, the particle may reset to a given position x r with a resetting rate r ∈ [0, 1] after each step of moving.…”

“…In [32,33,35], the authors treat resetting as an instantaneous action which means return to a given point without costing of time. Besides in [36], a two-state model has been developed, where the two states respectively correspond to the phases of movement and returning to the origin at a constant velocity.…”