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

By Franccois Bolley, José A. Cañizo, José Antonio Carrillo. 2010

221 citations recorded by the source corpus.

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.

Subfields: Global and Planetary Change, Molecular Biology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0644f6d78be307e720d61a2cc84c152f9fd1185f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Global and Planetary Change

1. Pierre-Emmanuel Jabin
2. Zhenfu Wang
3. José Antonio Carrillo

In Molecular Biology

1. Pierre-Emmanuel Jabin
2. Young-Pil Choi
3. Zhenfu Wang
4. M. Hauray
5. José Antonio Carrillo
6. Michael Winkler
7. Jacob Bedrossian

In Statistical and Nonlinear Physics

1. Zhenfu Wang
2. Pierre-Emmanuel Jabin
3. Young-Pil Choi
4. José Antonio Carrillo
5. M. Hauray
6. Daniel Lacker

Citations in the corpus

• Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces (2015, background): “In the case ε = 0, we can easily propagate the bound for |∇ log f | by tracing back the characteristics.”
• The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, background): “These ideas were improved to allow for locally Lipschitz velocity fields for compactly supported initial measures in [22] and for suitable decay conditions at infinity and with noise in [16].”
• Mean Field Limit for Stochastic Particle Systems (2017, methodology): “We also refer to [7, 9, 12] and the reference therein for recent developments and to [64] for the quantitative Grunbaum’s duality method.”