Properties of additive functionals of Brownian motion with resetting

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506

600

In this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate r, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate r. We then generalise to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.

Section: Large Deviationsmentioning

confidence: 99%

Stochastic resetting and applications

Evans

Majumdar

Schehr

2020

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506

600

“…Hence, in the subsequent interval [∆t, t + ∆t], the particle starts either from x 0 (the former case) or from x + dx (the latter case). Taking into account all these possibilities, we have G r (x, y, k, t + ∆t) = r∆tG r (x 0 , y, k, t) + (1 − r∆t)e −kU (x)∆t G r (x + ∆x, y, k, t) ∆x , with U(x) = δ(x − y) , (7) where the expectation on the second term is taken on different noise realizations [33,34]. We note that the term exp(−kU(x)∆t) above is somewhat ambiguous since powers of a delta function are not well defined.…”

Section: Local Time Statistics: General Formulationmentioning

confidence: 99%

“…Using renewal properties of the resetting dynamics, they derived a formula which links generating functions with and without resetting; and then used this formula to, e.g., compute the large deviations in the area covered by an Ornstein-Uhlenbeck trajectory with resetting. In a followup study [34], the authors focused on additive functionals of Brownian motion with resetting. They derived a large deviation principle in the presence of resetting and identified the large deviation rate function in terms of a variational formula involving the large deviation rate functions without resetting.…”

Section: Introductionmentioning

confidence: 99%

Local time of diffusion with stochastic resetting

Pal¹,

Chatterjee²,

Reuveni³

et al. 2019

Diffusion with stochastic resetting has recently emerged as a powerful modeling tool with a myriad of potential applications. Here, we study local time in this model, covering situations of free and biased diffusion with, and without, the presence of an absorbing boundary. Given a Brownian trajectory that evolved for t units of time, the local time is simply defined as the total time the trajectory spent in a small vicinity of its initial position. However, as Brownian trajectories are stochastic -the local time itself is a random variable which fluctuates round and about its mean value. In the past, the statistics of these fluctuations has been quantified in detail; but not in the presence of resetting which biases the particle to spend more time near its starting point. Here, we extend past results to include the possibility of stochastic resetting with, and without, the presence of an absorbing boundary and/or drift. We obtain exact results for the moments and distribution of the local time and these reveal that its statistics usually admits a simple form in the long-time limit. And yet, while fluctuations in the absence of stochastic resetting are typically non-Gaussian -resetting gives rise to Gaussian fluctuations. The analytical findings presented herein are in excellent agreement with numerical simulations. arXiv:1902.00907v2 [cond-mat.stat-mech] 5 Jun 2019 2

“…Recent variations on the resetting theme have been to consider: resetting of discretetime Lévy flights [37] and continuous-time Lévy walks [38,39], resetting for random walks in a bounded domain [40,39], resetting of extended systems such as fluctuating interfaces [41] and a reaction diffusion process in one dimension [42], Michaelis-Menten reaction schemes [43,35], the thermodynamics of resetting [44,45] and large deviations of the additive functionals of resetting processes [46,47,48], interaction-driven resetting [49], resetting with branching [50] and fractional Brownian motion with resetting [51]. Very recently, resetting dynamics in quantum systems have also been studied [52,53].…”

Section: Introductionmentioning

confidence: 99%

Run and tumble particle under resetting: a renewal approach

Evans¹,

Majumdar²

2018

Self Cite

244

258

We consider a particle undergoing run and tumble dynamics, in which its velocity stochastically reverses, in one dimension. We study the addition of a Poissonian resetting process occurring with rate r. At a reset event the particle's position is returned to the resetting site X r and the particle's velocity is reversed with probability η. The case η = 1/2 corresponds to position resetting and velocity randomization whereas η = 0 corresponds to position-only resetting. We show that, beginning from symmetric initial conditions, the stationary state does not depend on η i.e. it is independent of the velocity resetting protocol. However, in the presence of an absorbing boundary at the origin, the survival probability and mean time to absorption do depend on the velocity resetting protocol. Using a renewal equation approach, we show that the mean time to absorption is always less for velocity randomization than for position-only resetting.