Nonlinear self-stabilizing processes – I Existence, invariant probability, propagation of chaos

Benachour, Saı̈d; Roynette, Bernard; Talay, Denis; Vallois, Pierre

doi:10.1016/s0304-4149(98)00018-0

Cited by 84 publications

(102 citation statements)

References 6 publications

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“…2. Let us note that the uniqueness of symmetric invariant measure was already studied in [1] where the authors considered the constant potential case V (x) := 0. They obtained uniqueness results for α large enough but fixed which is to relate to our situation where α > θ but the noise intensity should be small.…”

Section: Combining Propositions 52-54 Leads To the Statement Of Thementioning

confidence: 99%

“…In other words the function b introduced above satisfies b(x, y) := −V (x) − F (x − y): V is called the environment potential and F represents the interaction potential. The functions V and F are assumed to verify different conditions developed at the end of the introduction and related to [1,2]. Let us just note two principal properties: F is an even convex function with F (0) = 0 and lim x→∞ F (x) = +∞ and V is an even double-well potential whose global minima are reached for x = −a and x = a > 0.…”

Section: Introductionmentioning

confidence: 99%

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Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit

Herrmann

Tugaut

2012

ESAIM: PS

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Abstract. In the context of self-stabilizing processes, that is processes attracted by their own law, living in a potential landscape, we investigate different properties of the invariant measures. The interaction between the process and its law leads to nonlinear stochastic differential equations. In [S. Herrmann and J. Tugaut. Electron. J. Probab. 15 (2010Probab. 15 ( ) 2087Probab. 15 ( -2116, the authors proved that, for linear interaction and under suitable conditions, there exists a unique symmetric limit measure associated to the set of invariant measures in the small-noise limit. The aim of this study is essentially to point out that this statement leads to the existence, as the noise intensity is small, of one unique symmetric invariant measure for the self-stabilizing process. Informations about the asymmetric measures shall be presented too. The main key consists in estimating the convergence rate for sequences of stationary measures using generalized Laplace's method approximations.Mathematics Subject Classification. 60J60, 60H10, 41A60.

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Section: Combining Propositions 52-54 Leads To the Statement Of Thementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit

Herrmann

Tugaut

2012

ESAIM: PS

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“…We extend to R + by classical arguments. We refer to [BRTV98,HIP08]. Let us remark that for all T < ∞, we have…”

Section: Introductionmentioning

confidence: 99%

Phase transitions of McKean–Vlasov processes in double-wells landscape

Tugaut

2013

Stochastics

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We aim to establish results about a particular class of inhomogeneous processes, the so-called McKean-Vlasov diffusions. Such a diffusion corresponds to the hydrodynamical limit of an interacting particle system, the mean-field one. Existence and uniqueness of the invariant probability are classical results provided that the external force corresponds to the gradient of a convex potential. However, previous results, see , state that the non-convexity of this potential implies the nonuniqueness of the invariant probabilities under easily checked assumptions. Here, we prove that there exists phase transitions, that is under a critical value, there are exactly three invariant probabilities and over another critical value, there is exactly one. Under simple assumptions, these two critical values coincide and it is characterize by a simple implicit equation. We exhibit other cases in which phase transitions occur. Finally, we proceed numerical estimations of the critical values.

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“…Under regularity and growth assumptions on σ, b and c, and if the particles are initially distributed in a chaotic way, for instance as independent and identically distributed variables, thenμ N t indeed converges as N tends to infinity to the distribution μ t of X t . The convergence ofμ N t is strongly linked with the phenomenon of propagation of chaos for the N interacting particles (X i t ) t≥0 , as we shall see below more in detail, and both issues have been studied in [1,8,17,18] for instance. Then quantitative estimates on this convergence have been obtained in [16] at the level of observables, and at the very level of the law in [6].…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…With this inequality in hand, given 1 ≤ p < 2 and λ 2 < λ 1 < λ, one deduces from (3.5) the existence of positive constants δ 1 , η 1 and K 1 such that…”

Section: Sanov's Argument On Small Ballsmentioning

confidence: 99%

Quantitative concentration inequalities on sample path space for mean field interaction

Bolley

2010

ESAIM: PS

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Abstract.We consider the approximation of a mean field stochastic process by a large interacting particle system. We derive non-asymptotic large deviation bounds measuring the concentration of the empirical measure of the paths of the particles around the law of the process. The method is based on a coupling argument, strong integrability estimates on the paths in Hölder norm, and a general concentration result for the empirical measure of identically distributed independent paths. Mathematics Subject Classification. 82C22, 35K55, 90C08.Received April 3, 2008. Revised September 29, 2008 This paper is devoted to the study of the particle approximation of a mean field stochastic process. In the models to be considered, the evolution is governed by a random diffusive term, an exterior force field and a mean field interaction depending on the law of the process itself.Quantitative estimates on the approximation have been obtained at the level of the time marginals. Let indeed μ t be the law of the considered process at time t, and (X i t ) 1≤i≤N be the position of the N particles in the phase space R d ; let alsoμ N t denote their empirical measure. First, [16] adapted concentration of measure ideas to obtain non-asymptotic bounds on the deviation of observables 1where ϕ is a given Lipschitz function on R d . Then, transposing Sanov's large deviation argument to their setting, [6] got non-asymptotic bounds on the deviation ofμ N t around μ t at the very level of the measures, namely for a distance which induces a topology stronger that the narrow topology.In this work we go one step further again by considering the law μ [0,T ] of the paths of the process on a given time interval [0, T ]. A natural object to consider in the particle approximation is now the empirical measurê μ N [0,T ] of the N trajectories (X i t ) 0≤t≤T . We shall give a precise meaning and estimates on the convergence ofμ N [0,T ] to μ [0,T ] ; we shall see that they imply previously mentioned results by projection at time t, but above all they give concentration estimates at the level of the paths.In the first section we state our main results and give an insight of the proofs, which will be given in more detail in the following sections.

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Nonlinear self-stabilizing processes – I Existence, invariant probability, propagation of chaos

Cited by 84 publications

References 6 publications

Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit

Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit

Phase transitions of McKean–Vlasov processes in double-wells landscape

Quantitative concentration inequalities on sample path space for mean field interaction

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