Large deviation properties of weakly interacting processes via weak convergence methods

Budhiraja, Amarjit; Dupuis, Paul; Fischer, Markus

doi:10.1214/10-aop616

Cited by 94 publications

(146 citation statements)

References 33 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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“…In Section 5 we apply the result in Theorem 1 to various corollaries, including the particular case when µ is the solution of a martingale problem. We finish by comparing our results to those of [19] and [20].…”

Section: Outline Of Main Resultsmentioning

confidence: 93%

“…In the course of this section we make progressively stronger assumptions on the nature of µ, culminating in the elegant expression for R(µ||P ) when µ is a solution of a martingale problem. We finish by comparing our work with that of [19,20]. Corollary 1.…”

Section: Some Corollariesmentioning

confidence: 92%

“…Sanov's theorem dictates that the empirical measure induced by independent samples governed by the same probability law P converge towards their limit exponentially fast; and the constant governing the rate of convergence is the relative entropy [14]. Large Deviations have been applied for example to spin glasses [15], neural networks [16][17][18] and mean-field models of interacting particles [19,20]. In the mean-field theory of neuroscience in particular, there has been a recent interest in the modelling of "finite size effects" [18,21], that is, the deviations from the limiting behaviour for a population of a particular size.…”

Section: Below)mentioning

confidence: 99%

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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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“…In particular, a lot of work has been done on spin glass dynamics, including Ben Arous and Guionnet on the mathematical side [1][2][3][4] and Sompolinsky and his co-workers on the theoretical physics side [5][6][7][8]. Furthermore, the large deviations of weakly interacting diffusions has been extensively studied by Dawson and Gartner [9,10], and more recently Budhiraja, Dupuis and Fischer [11,12]. More references to previous work on this particular subject can be found in these references.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Description of Neural Networks with Correlated Synaptic Weights

Faugeras

MacLaurin

2015

Entropy

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Abstract:We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network when the number of neurons goes to infinity. We introduce the process-level empirical measure of the trajectories of the solutions to the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions to the equations of the network of neurons. The main result of this article is that the image law through the empirical measure satisfies a large deviation principle with a good rate function which is shown to have a unique global minimum. Our analysis of the rate function allows us also to characterize the limit measure as the image of a stationary Gaussian measure defined on a transformed set of trajectories.

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Large deviation properties of weakly interacting processes via weak convergence methods

Cited by 94 publications

References 33 publications

Variational approach to coarse-graining of generalized gradient flows

Variational approach to coarse-graining of generalized gradient flows

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

Asymptotic Description of Neural Networks with Correlated Synaptic Weights

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