Large deviations from the mckean-vlasov limit for weakly interacting diffusions

Dawsont, Donald A.; Gärtner, Jürgen

doi:10.1080/17442508708833446

Cited by 333 publications

(401 citation statements)

References 16 publications

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“…In particular it has been shown that the sequence (ρ n ) has a large-deviation property [9,22,26] which characterizes the probability of finding the empirical measure far from the limit ρ, written informally as…”

Section: Origin Of the Functional I ε : Large Deviations Of A Stochasmentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with nondissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the smallnoise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. Mathematics Subject Classification

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Section: Origin Of the Functional I ε : Large Deviations Of A Stochasmentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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“…Because this intensity is inverse proportional to N, we obtain the resonancelike curve of the response in dependence of the system size. The main idea behind the system size resonance is that in finite ensembles of noise-driven or chaotic systems the dynamics of the mean field can be represented as driven by the effective noise whose variance is inverse proportional to the system size [9]. This idea has been applied to the description of a transition to collective behavior in [10].…”

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confidence: 99%

System Size Resonance in Coupled Noisy Systems and in the Ising Model

2002

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We consider an ensemble of coupled nonlinear noisy oscillators demonstrating in the thermodynamic limit an Ising-type transition. In the ordered phase and for finite ensembles stochastic flips of the mean field are observed with the rate depending on the ensemble size. When a small periodic force acts on the ensemble, the linear response of the system has a maximum at a certain system size, similar to the stochastic resonance phenomenon. We demonstrate this effect of system size resonance for different types of noisy oscillators and for different ensembles -lattices with nearest neighbors coupling and globally coupled populations. The Ising model is also shown to demonstrate the system size resonance. DOI: 10.1103/PhysRevLett.88.050601 PACS numbers: 05.40.Ca, 05.45. -a, 05.50. +q Stochastic resonance has attracted much interest recently [1]. As was demonstrated in [2], a response of a noisy nonlinear system to a periodic forcing can exhibit a resonancelike dependence on the noise intensity. In other words, there exists a "resonant" noise intensity at which the response to a periodic force is maximally ordered. Stochastic resonance has been observed in numerous experiments [3]. Noteworthy, the order in a noise-driven system can have a maximum at a certain noise level even in the absence of periodic forcing, this phenomenon being called coherence resonance [4].Being first discussed in the context of a simple bistable model, stochastic resonance has been also studied in complex systems consisting of many elementary bistable cells [5]; moreover, the resonance may be enhanced due to coupling [6]. In this paper we discuss another type of resonance in such systems, namely, the system size resonance, when the dynamics is maximally ordered at a certain number of interacting subsystems. Contrary to previous reports of array-enhanced stochastic resonance (cf. also [7]), here we fix the noise strength, coupling, and other parameters; only the size of the ensemble changes.The basic model to be considered below is the ensemble of noise-driven bistable overdamped oscillators, governed by the Langevin equations,Here j i ͑t͒ is a Gaussian white noise with zero mean: ͗j i ͑t͒j j ͑t 0 ͒͘ d ij d͑t 2 t 0 ͒;´is the coupling constant; N is the number of elements in the ensemble, and f͑t͒ is a periodic force to be specified later. In the absence of periodic force, the model (1) has been extensively studied in the thermodynamic limit N !`. It demonstrates an Isingtype phase transition at´´c from the disordered state with vanishing mean field X N 21 P i x i to the "ferromagnetic" state with a nonzero mean field X 6X 0 [8].While in the thermodynamic limit the full description of the dynamics is possible, for finite system sizes we have mainly a qualitative picture: In the ordered phase the mean field X switches between the values 6X 0 and its average vanishes for all couplings. The rate of switchings depends on the system size and tends to zero as N !`.For us, the main importance is the fact that qualitatively the behavior of the mean...

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“…The representation is in terms of the infinitesimal generator of P . This P is the same as in [22] (Section 4). The representation makes use of regular conditional probabilities.…”

Section: Below)mentioning

confidence: 83%

A Representation of the Relative Entropy with Respect to a Diffusion Process in Terms of Its Infinitesimal Generator

Faugeras

MacLaurin

2014

Entropy

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In this paper we derive an integral (with respect to time) representation of the relative entropy (or Kullback-Leibler Divergence) R(µ||P ), where µ and P are measuresThe underlying measure P is a weak solution to a martingale problem with continuous coefficients. Our representation is in the form of an integral with respect to its infinitesimal generator. This representation is of use in statistical inference (particularly involving medical imaging). Since R(µ||P ) governs the exponential rate of convergence of the empirical measure (according to Sanov's theorem), this representation is also of use in the numerical and analytical investigation of finite-size effects in systems of interacting diffusions.

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Large deviations from the mckean-vlasov limit for weakly interacting diffusions

Cited by 333 publications

References 16 publications

Variational approach to coarse-graining of generalized gradient flows

Variational approach to coarse-graining of generalized gradient flows

System Size Resonance in Coupled Noisy Systems and in the Ising Model

A Representation of the Relative Entropy with Respect to a Diffusion Process in Terms of Its Infinitesimal Generator

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