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

Christian Olivera, Marielle Simon

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.

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.
Paper Structure (15 sections, 13 theorems, 107 equations)

This paper contains 15 sections, 13 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Main results and examples of applications
  3. Some notations
  4. Setting and hypothesis
  5. Statement of the results
  6. Examples of applications
  7. Preliminaries
  8. Symmetric stable Lévy processes
  9. Maximal function and semi-group
  10. The equation for $u_{t}^{N}$ in mild form
  11. Proof of Theorem \ref{['th:rate2sing']}
  12. Proof of Corollary \ref{['Colo']}
  13. Intermediate lemmas
  14. Uniform bounds for $u^{N}$: proof of Lemma \ref{['estimation']}
  15. Martingale estimates

Key Result

Proposition 2.5

Given $u_0\in\mathcal{X}$ and $T>0$, there exists a unique mild solution $u$ to the initial-valued PDE problem eq:PDE on $[0,T]$. Moreover, if $v \in C([0,T];\mathcal{X})$ satisfies the integal equation eq:mildKSbis , with the constant $M$ verifying $M \geqslant \mathbf{C}_K \sup_{t\in [0,T]} \| u_

Theorems & Definitions (18)

  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Remark 2.8
  • Corollary 2.9
  • Definition 3.1: Lévy process
  • Proposition 3.2
  • ...and 8 more