  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 L\'evy
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.
