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Probabilistic Approach for Granular Media Equations in the Non Uniformly Convex Case
, 2007
"... We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of CarrilloMcCannVil ..."
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Cited by 14 (4 self)
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We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of CarrilloMcCannVillani [CMCV03, CMCV06] and completing results of Malrieu [Mal03] in the uniformly convex case. It relies on an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a T1 transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.
A PHASE TRANSITION BEHAVIOR FOR BROWNIAN MOTIONS INTERACTING THROUGH THEIR RANKS
"... Abstract. Consider a timevarying collection of n points on the positive real axis, modeled as exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their s ..."
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Cited by 13 (1 self)
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Abstract. Consider a timevarying collection of n points on the positive real axis, modeled as exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as n tends to infinity. Under a certain ‘continuity at the edge ’ condition on the drifts, we show that one of the following must happen: either (i) all points converge to 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a nontrivial PoissonDirichlet distribution. The proof employs, among other things, techniques from Talagrand’s analysis of the low temperature phase of Derrida’s Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the PoissonDirichlet law. 1.
Malrieu Trend to Equilibrium and Particle Approximation for a Weakly Self–Consistent VlasovFokkerPlanck Equation
, 2009
"... We consider a VlasovFokkerPlanck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. ..."
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Cited by 9 (3 self)
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We consider a VlasovFokkerPlanck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality for the distribution of the particle system leads to quantitative deviation bounds on the approximation of the equilibrium solution of the equation by an empirical mean of the particles at given time. Introduction and main results We are interested in the long time behaviour and in a particle approximation of a distribution ft(x,v) in the space of positions x ∈ R d and velocities v ∈ R d (with d � 1) evolving according to the VlasovFokkerPlanck equation
Stochastic meanfield limit: nonLipschitz forces and swarming
 Math. Models Methods Appl. Sci
"... We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the meanfield limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the l ..."
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Cited by 8 (1 self)
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We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the meanfield limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the literature such as the CuckerSmale model, adding noise to the behavior of individuals. The difficulty, as compared to the classical case of globally Lipschitz potentials, is that in several models the interaction potential between particles is only locally Lipschitz, the local Lipschitz constant growing to infinity with the size of the region considered. With this in mind, we present an extension of the classical theory for globally Lipschitz interactions, which works for only locally Lipschitz ones.
Large deviations and a Kramers’ type law for selfstabilizing diffusions, in "Ann
 Appl. Probab
"... We investigate exit times from domains of attraction for the motion of a selfstabilized particle traveling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Selfstabilization is the effect of including an ensembleaverage attraction in addition to the us ..."
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Cited by 6 (0 self)
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We investigate exit times from domains of attraction for the motion of a selfstabilized particle traveling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Selfstabilization is the effect of including an ensembleaverage attraction in addition to the usual statedependent drift, where the particle is supposed to be suspended in a large population of identical ones. A Kramers ’ type law for the particle’s exit from the potential’s domains of attraction and a large deviations principle for the selfstabilizing diffusion are proved. It turns out that the exit law for the selfstabilizing diffusion coincides with the exit law of a potential diffusion without selfstabilization and a drift component perturbed by average attraction. We show that selfstabilization may substantially delay the exit from domains of attraction, and that the exit location may be completely different. 1. Introduction. We
Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf
, 2007
"... In this paper, in the particular case of a concave flux function, we are interested in the long time behavior of the nonlinear process associated in [Methodol. Comput. Appl. Probab. 2 (2000) 69–91] to the onedimensional viscous scalar conservation law. We also consider the particle system obtained ..."
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Cited by 6 (3 self)
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In this paper, in the particular case of a concave flux function, we are interested in the long time behavior of the nonlinear process associated in [Methodol. Comput. Appl. Probab. 2 (2000) 69–91] to the onedimensional viscous scalar conservation law. We also consider the particle system obtained by replacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cumulative distribution function. We first obtain a trajectorial propagation of chaos estimate which strengthens the weak convergence result obtained in [8] without any convexity assumption on the flux function. Then Poincaré inequalities are used to get explicit estimates concerning the long time behavior of both the nonlinear process and the particle system. Introduction. In this paper, we are interested in the viscous scalar conservation law with C 1 flux function −A (1)
The McKean –Vlasov Equation in Finite Volume
, 2009
"... Abstract: We study the McKean–Vlasov equation on the finite tori of length scale L in d–dimensions. We (re)derive the necessary and sufficient conditions for the existence of a phase transition – first uncovered in [13] and [20]. Therein and in subsequent works, one finds indications pointing to cri ..."
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Cited by 4 (1 self)
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Abstract: We study the McKean–Vlasov equation on the finite tori of length scale L in d–dimensions. We (re)derive the necessary and sufficient conditions for the existence of a phase transition – first uncovered in [13] and [20]. Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, θ ♯ of the interaction parameter. We show (there is a basin of) dynamical stability for θ < θ ♯ and prove, abstractly, that a critical transition must occur at θ = θ ♯. However for this system we show that under generic conditions – L large, d ≥ 2 and isotropic interactions – the phase transition is in fact discontinuous and occurs at some θT < θ ♯. Finally, for H–stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the θT(L) tend to a definitive non–trivial limit as L → ∞. 1
DOI 10.1007/s109550099913z The McKean–Vlasov Equation in Finite Volume
"... Abstract We study the McKean–Vlasov equation on the finite tori of length scale L in ddimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in Gates and Penrose (Commun. Math. Phys. 17:194–209, 1970) an ..."
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Abstract We study the McKean–Vlasov equation on the finite tori of length scale L in ddimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in Gates and Penrose (Commun. Math. Phys. 17:194–209, 1970) and Kirkwood and Monroe (J. Chem. Phys. 9:514–526, 1941). Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, θ ♯ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for θ<θ ♯ and prove, abstractly, that a critical transition must occur at θ = θ ♯.Howeverforthis system we show that under generic conditions—L large, d ≥ 2 and isotropic interactions— the phase transition is in fact discontinuous and occurs at some θT <θ ♯. Finally, for Hstable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the θT(L) tend to a definitive nontrivial limit as L →∞.