Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp propagation of chaos for McKean-Vlasov equation with non constant diffusion coefficient

Jules Grass, Arnaud Guillin, Christophe Poquet

Abstract

We present a method to obtain sharp local propagation of chaos results for a system of N particles with a diffusion coefficient that it not constant and may depend of the empirical measure. This extends the recent works of Lacker [14] and Wang [24] to the case of non constant diffusions. The proof relies on the BBGKY hierarchy to obtain a system of differential inequalities on the relative entropy of k particles, involving the fisher information.

Sharp propagation of chaos for McKean-Vlasov equation with non constant diffusion coefficient

Abstract

We present a method to obtain sharp local propagation of chaos results for a system of N particles with a diffusion coefficient that it not constant and may depend of the empirical measure. This extends the recent works of Lacker [14] and Wang [24] to the case of non constant diffusions. The proof relies on the BBGKY hierarchy to obtain a system of differential inequalities on the relative entropy of k particles, involving the fisher information.

Paper Structure

This paper contains 4 sections, 2 theorems, 48 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of theorem \ref{['th main']}
  3. Computation of the relative entropy and derivation of the system of differential inequalities
  4. Estimation of the system of inequalities

Key Result

Theorem 1.1

Let $\bigl(V_{t}^{i,N} \bigl)_{i=1,..,N}$ solving ((1)). Let us assume that $b$, $\nabla \cdot a_{1}$, $a_{2}$ and $\nabla \cdot a_{2}$ are bounded. Suppose also the following: Then, for all $t>0$ there exists a constant $M_t$ independent of $N$ and $k$ such that

Theorems & Definitions (2)

  • Theorem 1.1
  • Proposition 2.1