Local large deviation principle for Wiener process with random resetting

Logachov, A. V.; Logachova, O.; Yambartsev, Anatoly

doi:10.1142/s021949372050032x

Cited by 4 publications

(6 citation statements)

References 26 publications

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“…We remind that N t := sup{n ≥ 0 : T n ≤ t} is the number of renewals by the time t. We have Z t = B n,t−Tn−1 for t ∈ [T n−1 , T n ) and n ≥ 1, so that Z t is constructed by pasting together fresh Brownian motions starting from the origin at each renewal time. We assume that the Brownian motions are independent of the waiting times, according to the reference literature [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. An additive functional of the process {Z t } t≥0 is the random variable…”

Section: Brownian Motion Under Resetting and Fluctuationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Physicists have focused much of their work on versions of the Brownian motion under Poissonian resetting, addressing deviations from the typical behavior [4][5][6][7][8] and first-passage time problems in connection with intermittent search strategies [9][10][11][12][13][14][15][16][17][18]. Regarding the former, the ability of stochastic resetting to shape fluctuations has been tested for additive functionals of the process, such as the positive occupation time [5], the area [4,5], the absolute area [5], and the local time [6,7], as well as positive and negative excursions [8]. As resetting has the effect of confining the process around the initial position, leading to the emergence of a stationary state for these variants of the Brownian motion [2], one natural question is whether or not the resetting mechanism can bring about a large deviation principle for an additive functional when it does not satisfy a large deviation principle without resetting.…”

Section: Introductionmentioning

confidence: 99%

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Statistical fluctuations under resetting: rigorous results

Zamparo¹

2022

Preprint

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In this paper we investigate the normal and the large fluctuations of additive functionals of the Brownian motion under a general non-Poissonian resetting mechanism. Cumulative functionals of regenerative processes are very close to renewal-reward processes and inherit most of the properties of the latter. Here we review and use the classical law of large numbers and central limit theorem for renewal-reward processes to obtain same theorems for additive functionals of the reset Brownian motion. Then, we establish large deviation principles for these functionals by illustrating and applying a large deviation theory for renewal-reward processes that has been recently developed by the author. We discuss applications of the general results to the positive occupation time, the area, and the absolute area. While introducing advanced tools from renewal theory, we demonstrate that a rich phenomenology accounting for dynamical phase transitions emerges when one goes beyond Poissonian resetting.

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Section: Brownian Motion Under Resetting and Fluctuationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Statistical fluctuations under resetting: rigorous results

Zamparo¹

2022

Preprint

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“…Furthermore, some papers also regard the resetting position x r as a random variable. In [9,[60][61][62], the value of x r is drawn from a distribution that does not change with time. To the best of our knowledge, [63,64] are the first (and, up until now, the only) ones that deal with the case where the distribution of x r is time-dependent.…”

Section: Introductionmentioning

confidence: 99%

On a diffusion which stochastically restarts from moving random spatial positions: a non-renewal framework

da Silva

2023

J. Phys. A: Math. Theor.

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We consider a diffusive particle that at random times, exponentially distributed with parameter β, stops its motion and restarts from a moving random position Y(t) in space. The position X(t) of the particle and the restarts do not affect the dynamics of Y(t), so our framework constitutes in a non-renewal one. We exhibit the feasibility to build a rigorous general theory in this setup from the analysis of sample paths. To prove the stochastic process X(t) has a non-equilibrium steady-state, assumptions related to the confinement of Y(t) have to be imposed. In addition we design a detailed example where the random restart positions are provided by the paradigmatic Evans and Majumdar’s diffusion with stochastic resettings (Evans M and Majumdar S 2011 Phys. Rev. Lett. 106 160601), with resetting rate β Y . We show the ergodic property for the main process and for the stochastic process of jumps performed by the particle. A striking feature emerges from the examination of the jumps, since their negative covariance can be minimized with respect to both rates β and β Y , independently. Moreover we discuss the theoretical consequences that this non-renewal model entails for the analytical study of the mean first-passage time (FPT) and mean cost up to FPT.

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“…In particular, the ability of stochastic resetting to shape fluctuations has been tested for additive functionals, such as the positive occupation time of the Brownian motion [4], its area and absolute area [4], the area of the fractional Brownian motion [5], the area of the Ornstein-Uhlenbeck process [6], and the local time of the Brownian motion [7,8]. For the Brownian motion subjected to Poissonian resetting, also positive and negative excursions have been investigated [9]. As resetting has the effect of confining the process around the initial position, leading to the emergence of a stationary state for the reset Brownian motion [2], one natural question is whether or not the resetting mechanism can bring about a large deviation principle for an additive functional when it does not satisfy a large deviation principle without resetting.…”

Section: Introductionmentioning

confidence: 99%

Statistical fluctuations under resetting: rigorous results

Zamparo¹

2022

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

In this paper we investigate the normal and the large fluctuations of additive functionals associated with a stochastic process under a general non-Poissonian resetting mechanism. Cumulative functionals of regenerative processes are very close to renewal-reward processes and inherit most of the properties of the latter. Here we review and use the classical law of large numbers and central limit theorem for renewal-reward processes to obtain same theorems for additive functionals of a stochastic process under resetting. Then, we establish large deviation principles for these functionals by illustrating and applying a large deviation theory for renewal-reward processes that has been recently developed by the author. We discuss applications of the general results to the positive occupation time, the area, and the absolute area of the reset Brownian motion. While introducing advanced tools from renewal theory, we demonstrate that a rich phenomenology accounting for dynamical phase transitions emerges when one goes beyond Poissonian resetting.

show abstract

Local large deviation principle for Wiener process with random resetting

Cited by 4 publications

References 26 publications

Statistical fluctuations under resetting: rigorous results

Statistical fluctuations under resetting: rigorous results

On a diffusion which stochastically restarts from moving random spatial positions: a non-renewal framework

Statistical fluctuations under resetting: rigorous results

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