Microscopic derivation of non-local models with anomalous diffusions from stochastic particle systems

Abstract

This paper considers a large class of nonlinear integro-differential scalar equations which involve an anomalous diffusion (e.g. driven by a fractional Laplacian) and a non-local singular convolution kernel. Each of those singular equations is obtained as the macroscopic limit of an interacting particle system modeled as $N$ coupled stochastic differential equations driven by Lévy processes. In particular we derive quantitative estimates between the microscopic empirical measure of the particle system and the solution to the limit equation in some non-homogeneous Sobolev space. Our result only requires very weak regularity on the interaction kernel, therefore it includes numerous applications, e.g.: the $2d$ turbulence model (including the quasi-geostrophic equation) in sub-critical regime, the $2d$ generalized Navier–Stokes equation, the fractional Keller–Segel equation in any dimension, and the fractal Burgers equation.