Convergence to equilibrium for granular media equations and their Euler schemes

Malrieu, Florent

doi:10.1214/aoap/1050689593

Cited by 120 publications

(155 citation statements)

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“…See [Szn91] under Lipschitz properties, [BRTV98] if V is a constant, [Mal01,Mal03] when both potentials are convex. Under some convexity properties, this coupling (which is equivalent to the propagation of chaos) holds on R + , see [CGM08].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…(I) or the associated mean-field system of particles, see [CGM08,Fun84,Mal03]. Reciprocally, Equation (II) is a useful tool to describe the invariant probabilities and the long-time behavior, see [BRTV98,BRV98,Tam84,Tam87,Ver06].…”

Section: Introductionmentioning

confidence: 99%

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

Tugaut

2013

Stochastics

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“…On the implicit Euler approximation in stochastic analysis, cf. F. Malrieu [15] and, F. Malrieu and D. Talay [16] for example.…”

Section: 3mentioning

confidence: 99%

A generalized mean-reverting equation and applications

Marie

2014

ESAIM: PS

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Abstract. Consider a mean-reverting equation, generalized in the sense it is driven by a 1-dimensional centered Gaussian process with Hölder continuous paths on [0, T ] (T > 0). Taking that equation in rough paths sense only gives local existence of the solution because the non-explosion condition is not satisfied in general. Under natural assumptions, by using specific methods, we show the global existence and uniqueness of the solution, its integrability, the continuity and differentiability of the associated Itô map, and we provide an L p -converging approximation with a rate of convergence (p 1). The regularity of the Itô map ensures a large deviation principle, and the existence of a density with respect to Lebesgue's measure, for the solution of that generalized mean-reverting equation. Finally, we study a generalized mean-reverting pharmacokinetic model.

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“…the Monge transport norm (see [8,5]) is defined as [7] for the elliptic operator L; let ρ S be the measure obtained by the scheme S with initial value Z W ε (0) = ξ 0 . Then…”

Section: Definition Of the Scheme Smentioning

confidence: 99%