Derivation of mean-field equations for stochastic particle systems

Großkinsky, Stefan; Jatuviriyapornchai, Watthanan

doi:10.1016/j.spa.2018.05.006

Cited by 10 publications

(7 citation statements)

References 45 publications

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“…Similar laws of large numbers for empirical measures of particle systems can be found for a huge class of processes in the literature, e.g. [25].…”

Section: Introductionsupporting

confidence: 73%

Continuum and thermodynamic limits for a simple random-exchange model

Düring¹,

Georgiou²,

Merino-Aceituno³

et al. 2020

Preprint

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We discuss various limits of a simple random exchange model that can be used for the distribution of wealth. We start from a discrete state space -discrete time version of this model and, under suitable scaling, we show its functional convergence to a continuous space -discrete time model. Then, we show a thermodynamic limit of the empirical distribution to the solution of a kinetic equation of Boltzmann type. We solve this equation and we show that the solutions coincide with the appropriate limits of the invariant measure for the Markov chain. In this way we complete Boltzmann's program of deriving kinetic equations from random dynamics for this simple model. Three families of invariant measures for the mean field limit are discovered and we show that only two of those families can be obtained as limits of the discrete system and the third is extraneous. Finally, we cast our results in the framework of integer partitions and strengthen some results already available in the literature.

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“…Similar laws of large numbers for empirical measures of particle systems can be found for a huge class of processes in the literature, e.g. [25].…”

Section: Introductionsupporting

confidence: 73%

Continuum and thermodynamic limits for a simple random-exchange model

Düring¹,

Georgiou²,

Merino-Aceituno³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The factor 1/N results from a self-averaging property of the mean-field interaction through selection dynamics, which is expected from results on other mean-field particle systems (see e.g. [35,36] and references therein), and is fully analogous to the central-limit type scaling of the empirical variance for the sum of N independent random variables. While this scaling remains the same for more general particle approximations with more than one particle being affected by selection events, the simple identity (3.16) does not hold exactly for any N ≥ 1 as we see in the next subsection.…”

Section: Essential Properties Of Particle Approximationsmentioning

confidence: 63%

Rare Event Simulation for Stochastic Dynamics in Continuous Time

et al. 2019

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Large deviations for additive path functionals of stochastic dynamics and related numerical approaches have attracted significant recent research interest. We focus on the question of convergence properties for cloning algorithms in continuous time, and establish connections to the literature of particle filters and sequential Monte Carlo methods. This enables us to derive rigorous convergence bounds for cloning algorithms which we report in this paper, with details of proofs given in a further publication. The tilted generator characterizing the large deviation rate function can be associated to non-linear processes which give rise to several representations of the dynamics and additional freedom for associated numerical approximations. We discuss these choices in detail, and combine insights from the filtering literature and cloning algorithms to compare different approaches and improve efficiency.

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“…In this section, our method is perturbative, for which reason we restrict to the complete graph, that is p(x, y) = 1 for all x = y. In this case, the mean-field limit from the N -particle system was derived in Grosskinsky and Jatuviriyapornchai (2019), and the limit equation was investigated in Schlichting (2019). Since positive curvature is know in the case of independent particles on the complete graph (Erbar and Maas, 2012), we expect that for c(µ x , µ y ) = T +c(µ x , µ y ) with bounded c : P(X ) × P(X ) → [0, ∞), we should also obtain positive entropic curvature for the non-linear models when T is sufficiently large.…”

Section: 2mentioning

confidence: 99%

Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces

Erbar¹,

Fathi²,

Schlichting³

2020

ALEA

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We consider non-linear evolution equations arising from mean-field limits of particle systems on discrete spaces. We investigate a notion of curvature bounds for these dynamics based on the convexity of the free energy along interpolations in a discrete transportation distance related to the gradient flow structure of the dynamics. This notion extends the one for linear Markov chain dynamics studied in Erbar and Maas (2012). We show that positive curvature bounds entail several functional inequalities controlling the convergence to equilibrium of the dynamics. We establish explicit curvature bounds for several examples of mean-field limits of various classical models from statistical mechanics.

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Derivation of mean-field equations for stochastic particle systems

Cited by 10 publications

References 45 publications

Continuum and thermodynamic limits for a simple random-exchange model

Continuum and thermodynamic limits for a simple random-exchange model

Rare Event Simulation for Stochastic Dynamics in Continuous Time

Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces

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