Stochastic Synchronization in a Large System of Identical Particles

Manita, Anatoly

doi:10.1137/s0040585x97983444

Cited by 12 publications

(18 citation statements)

References 3 publications

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“…To get this representation we need a chain of auxiliary results on the free dynamics and the interaction (Lemmas 4-6 in § 4.1-4.2). These lemmas are similar to their analogues proved for Markovian models in [46,47]. Nevertheless, the proof of Lemma 6 meets additional difficulties related to the involved nature of the sequence of synchronization epochs T .…”

Section: Introductionsupporting

confidence: 61%

“…In this paper we consider stochastic synchronization systems which are essentially more general than many previously studied mathematical models [46,47,49,52]. For instance, the paper [47] studies a symmetric system of N identical Brownian particles with pairwise synchronization.…”

Section: Introductionmentioning

confidence: 99%

“…The first rigorous treatment of a multi-dimensional mathematical model with time stamp synchronization was done in [43]. In [41,43,46,47,49,52] different N-component synchronization systems were considered as stochastic particle systems with special interaction. Such interpretation is useful for invoking physical intuition.…”

Section: Introductionmentioning

confidence: 99%

“…The problem is to find different time scales (t = t N → ∞ as N → ∞) on which the synchronization system x(t N ) demonstrates completely different qualitative behaviour. The complete description of times scales was obtained for several models [40,[46][47][48]51]. For example, in [47] it was shown that the "BM N -model" passes three different phases before it reaches the final synchronization.…”

Section: Introductionmentioning

confidence: 99%

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Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

Manita

2014

Preprint

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We propose stochastic N -component synchronization models (x 1 (t), . . . , x N (t)), x j ∈ R d , t ∈ R + , whose dynamics is described by Lévy processes and synchronizing jumps. We prove that symmetric models reach synchronization in a stochastic sense: differences between components d (N ) kj (t) = x k (t)−x j (t) have limits in distribution as t → ∞. We give conditions of existence of natural (intrinsic) space scales for large synchronized systems, i.e., we are looking for such sequences {b N } that distribution of d (N ) kj (∞)/b N converges to some limit as N → ∞. It appears that such sequence exists if the Lévy process enters a domain of attraction of some stable law. For Markovian synchronization models based on α-stable Lévy processes this results holds for any finite N in the precise form with b N = (N − 1) 1/α . For non-Markovian models similar results hold only in the asymptotic sense. The class of limiting laws includes the Linnik distributions. We also discuss generalizations of these theorems to the case of non-uniform matrix-based intrinsic scales. The central point of our proofs is a representation of characteristic functions of d (N ) kj (t) via probability distribution of a superposition of N independent renewal processes.

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Section: Introductionsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

Manita

2014

Preprint

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“…Subsequently to [5,6], there has been a line of work generally focused on modeling synchronization between elements (particles) of a large system, with mean-field interaction of the elements (particles), cf. [9][10][11][12][13][14].…”

Section: Other Related Workmentioning

confidence: 99%

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Stolyar¹

2020

Preprint

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We use probabilistic methods to study properties of mean-field models, arising as large-scale limits of certain particle systems with mean-field interaction. The underlying particle system is such that n particles move forward on the real line. Specifically, each particle "jumps forward" at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle's location quantile within the overall distribution of particle locations. A mean-field model describes the evolution of the particles' distribution, when n is large. It is essentially a solution to an integro-differential equation within a certain class. Our main results concern the existence and uniqueness of -and attraction tomean-field models which are traveling waves, under general conditions on the jump-rate function and the jump-size distribution.

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Clock synchronization in symmetric stochastic networks

Manita

2014

Queueing Syst

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Stochastic Synchronization in a Large System of Identical Particles

Cited by 12 publications

References 3 publications

Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Clock synchronization in symmetric stochastic networks

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