Continuous-time random walks with reset events

Abstract

In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.

Spectral properties of simple classical and quantum reset processes

“…These studies follow many previous works in mathematics, in queuing theory and in population dynamics, in particular, on stochastic processes involving some form of random resets, variously referred to as failures, catastrophes, disasters or decimations; see, e.g., [17][18][19][20][21][22][23][24][25]. Most of these works, as well as those from physics mentioned above, make use of the correspondence that exists between resets and renewals to obtain renewal representations of both time-dependent and stationary distributions, in addition to first-passage statistics.…”

Anomalous Diffusion in Random-Walks With Memory-Induced Relocations

“…For example, as a consequence of such resetting, dispersal is asymptotically suppressed and a steady state is reached. 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.…”

Properties of additive functionals of Brownian motion with resetting

“…The properties of rBM, and reset processes in general [29], have been the subject of several recent studies, related to random searches and randomized algorithms [2-4, 6, 8, 10, 12, 19, 23] (which can be made more efficient by the addition of resetting [11]), queueing theory (where resetting accounts for the accidental clearing of queues or buffers), as well as birth-death processes [7,9,20,26,30,31] (in which a population is drastically reduced as a result of natural disasters or catastrophes). In biology, the attachment, targeting and transcription dynamics of enzymes, proteins and other bio-molecules can also be modelled with reset processes [5,15,27,[32][33][34]39].…”