Probabilistic approach for granular media equations in the non-uniformly convex case

By Patrick Cattiaux, Guillin Arnaud, Malrieu Florent. 2006

190 citations recorded by the source corpus.

We use here a particle system to prove both a convergence result (with convergence rate) and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. The proof of convergence is simpler than the one in Carrillo–McCann–Villani (Rev. Mat. Iberoamericana 19:971–1018, 2003; Arch. Rat. Mech. Anal. 179:217–263, 2006). All the results complete former results of Malrieu (Ann. Appl. Probab. 13:540–560, 2003) in the uniformly convex case. The main tool is 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.

Subfields: Applied Mathematics, Environmental Engineering

View Paper: https://www.semanticscholar.org/paper/0bbde999cb0dd33021af5669d72d8d11f256fdcb

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Citations in the corpus

Freidlin–Wentzell LDP in path space for McKean–Vlasov equations and the functional iterated logarithm law (2017, background): “4) with same law when the size of the system N goes to infinity, see [37] under Lipschitz properties; [30] under Lipschitz assumptions but allowing for jumps; [4] if V is a constant; [10] for a uniform result in time in the non-uniformly convex case.”
On the rate of convergence in Wasserstein distance of the empirical measure (2013, unknown): “This is a probabilistic representation of the so-called Mc Kean-Vlasov equation, which has been studied in particular by Carillo-Mac Cann-Villani [12], Malrieu [28] and Cattiaux-Guillin-Malrieu [13] to which we refer for further motivations and existence and uniqueness of solutions.”
Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes (2016, methodology): “6 below (where we will also assume that θ is Lipschitz) using [11,57], existence and uniqueness of the solutions hold.”
Semi Log-Concave Markov Diffusions (2013, background): “This is what is done in [37, 19] to study the long time behavior of Xt.”
Quantitative estimates of propagation of chaos for stochastic systems with W-1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d (2017, methodology): “as in the classical result [68]; see also [3,11,20,67] for more on the classical Lipschitz case.”
Mean Field Limit for Stochastic Particle Systems (2017, background): “The results referred to here were mostly obtained in [9, 10, 13, 57, 58].”
On Ergodic Properties of Nonlinear Markov Chains and Stochastic McKean--Vlasov Equations (2013, background): “These processes arise naturally in the study of the limit behavior of a large number of weakly interacting Markov processes ([2], [5], [19]) and have a wide range of applications, including financial mathematics, population dynamics, neuroscience (see, e.”
Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming (2010, methodology): “For space-homogeneous kinetic models it was extended to non-Lipschitz drifts by means of convexity arguments, first in one dimension in [4], then more generally in any dimension in [13, 27].”