Propagation of chaos for a subcritical Keller–Segel model

Abstract

This paper deals with a subcritical Keller-Segel equation. Starting from the stochastic particle system associated with it, we show well-posedness results and the propagation of chaos property. More precisely, we show that the empirical measure of the system tends towards the unique solution of the limit equation as the number of particles goes to infinity.Résumé. Cet article traite de l'équation de Keller-Segel dans un cadre sous-critique. À l'aide du système de particules en lien avec cette équation, nous montrons des résultats d'existence et d'unicité, puis la propagation du chaos pour ce dernier. Plus précisément, nous montrons que la mesure empirique du système tend vers l'unique solution de l'équation limite lorsque le nombre de particules tend vers l'infini. MSC: 65C35where f : R + ×R 2 → R and χ > 0. The force field kernel K : R 2 → R 2 comes from an attractive potential : R 2 → R and is defined by, α ∈ (0, 1).(1.2)The standard Keller-Segel equation correspond to the critical case K(x) = x/|x| 2 (i.e., more singular at x = 0) and it describes a model of chemotaxis, i.e., the movement of cells (usually bacteria or amoebae) which are attracted 966 D. Godinho and C. Quiñinao by some chemical substance that they produce. This equation has been first introduced by Keller and Segel in [15,16]. Blanchet, Dolbeault and Perthame showed in [4] some nice results on existence of global weak solutions if the nonnegative parameter χ (which is the sensitivity of the bacteria to the chemo-attractant) is smaller than 8π/M where M is the initial mass (here M will always be 1 since we will deal with probability measures). For more details on the subject, see [12,13].

A Microscopic Spiking Neuronal Network for the Age-Structured Model

“…then there exists a unique càdlàg adapted R 2 + -valued process (X N t , M N t ) t≥0 solution to (4)- (5).…”

“…A combined PDE/SDE approach to find a sharper entropy function could be useful to prove the path-wise uniqueness result and then to improve our results (see e.g. Godinho [5]). The second novelty of the present work is the presence of delays, that is central in the original PPS model.…”

“…So far, we know that, for each N ≥ 1 and (g 0 ⊗ m 0 )-chaotic initial states, there exists a unique solution to (4)- (5). We now study the convergence of the this set of solutions as N goes to infinity.…”

“…where the initial conditions and the Poisson measures are exactly the ones driving the particle system equations (4)- (5). We use the shorthand notation η N (t) for the empirical mean associated to the exchangeable family (X i t ,M i t ), and introduce the Monge-KantorovichWasserstein distance between two probability measures μ and η on R + by…”

Propagation of chaos for large Brownian particle system with Coulomb interaction

“…The sign of F is crucially used in Lemmas 2.2 and 2.3 to achieve the uniform estimates. For a related work, Godinho and Quininao proved propagation of chaos for the subcritical Keller-Segel equations [6]. Some of their frameworks and techniques will be adapted to this paper.…”