On mean-field limits and quantitative estimates with a large class of singular kernels: Application to the Patlak–Keller–Segel model

By Didier Bresch, Pierre-Emmanuel Jabin, Zhenfu Wang. Comptes rendus. Mathematique, 2019

91 citations recorded by the source corpus.

Abstract In this note, we propose a modulated free energy combination of the methods developed by P.-E. Jabin and Z. Wang [Inventiones (2018)] and by S. Serfaty [Proc. Int. Cong. Math. (2018) and references therein] to treat more general kernels in mean-field limit theory. This modulated free energy may be understood as introducing appropriate weights in the relative entropy developed by P.-E. Jabin and Z. Wang (in the spirit of what has been recently developed by D. Bresch and P.-E. Jabin [Ann. of Math. (2) (2018)]) to cancel the most singular terms involving the divergence of the flow. Our modulated free energy allows us to treat singular potentials that combine large smooth part, small attractive singular part, and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as the Patlak–Keller–Segel system in subcritical regimes, is obtained.

Subfields: Finance, Modeling and Simulation, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/63595f170665d597777428d55a1499854d9f0d4d

How researchers see this paper

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

In Finance

In Modeling and Simulation

In Statistics and Probability

Citations in the corpus

Distribution dependent SDEs driven by additive fractional Brownian motion (2021, methodology): “On the one hand, new methods based on entropy inequalities [6, 14, 28] and modulated energy methods [41, 42] have allowed for the rigorous derivation of mean field limits for fairly singular B; while on the other, DDSDEs with irregular drifts are related to the flourishing field of regularization by”
Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions (2021, background): “It is natural to wonder, though, if our general strategy of comparing the k-particle “energy” to the (k+1)-particle “energy” via the BBGKY hierarchy could be adapted for different kinds of dynamics and perhaps better-tailored notions of “energies,” rather than entropy, such as the modulated (free) e”