Stochastic Mean-Field Limit: Non-Lipschitz Forces and Swarming

Abstract

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 mean-field 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 Cucker-Smale 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.

A Microscopic Spiking Neuronal Network for the Age-Structured Model

“…Let f = f (t, x) ≥ 0 be the density of neurons in the state x ∈ R + at time t ≥ 0. The dynamics of the age-structured PPS model are given by the B C. Quiñinao cristobal.quininao@gmail.com 1 Departamento de Ecología, Facultad de Ciencias Biológicas, Universidad Católica, Avda. Libertador Bernardo O'Higgins 340, Santiago, Chile 2 Universidad de Tarapacá, Instituto de Alta Investigación, Antofagasta No.…”

“…, N} of interacting neurons. 1 The state of each neuron i is described by a nonnegative-valued variable X i,N t called age, corresponding to the time elapsed since last discharge. This approach is quite different from classical literature, where the key variable is the voltage, and constitutes an important originality of the PPS model.…”

Existence of Weak Solutions to Kinetic Flocking Models

“…There is a considerable body of literature concerning the kinetic Cucker-Smale equation and its variations (see [2,3,4,5,6,7,8,11,12,14]), but a general existence theory has thus far remained absent. Notable exceptions are the studies [1,5] in which well-posedness for Cucker-Smaletype models is established in the sense of measures and then extended to weak solutions ( [1] adds noise to the model). The existence of classical solutions to the kinetic Cucker-Smale equation is established in [12], but the result does not extend to (1.1).…”

Long time behaviour and particle approximation of a generalised Vlasov dynamic

“…We address both issues using the stochastic interpretation (3) of (1) and the so-called coupling technique introduced in [18,20] and used further in [6,5]. First, we prove the existence and uniqueness of a stationary solution as well as convergence to the stationary solution of other solutions of (1).…”

“…The underlying idea of the coupling technique introduced in [18,20] and used further in [6,5] is based on (14). The underlying idea of the coupling technique introduced in [18,20] and used further in [6,5] is based on (14).…”