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Uniform in Time Propagation of Chaos for Mean Field Particle System with Interacting Noise and Partially Dissipative Drifts

Xing Huang

TL;DR

This work addresses uniform-in-time propagation of chaos in $W_1$ for a mean-field particle system with interacting diffusion coefficients and a partially dissipative drift. The authors develop a framework based on reflection coupling, gradient estimates for time-inhomogeneous decoupled SDEs, and a Duhamel formula for two semigroups to handle the coupling between the $N$-particle system and the nonlinear limit. They prove an explicit exponential decay bound with a $N^{-1/2}$ residual: for all $t\ge0$ and $1\le k\le N$, $W_1((P_t)^k)^*\mu_0^N,(P_t^*\mu_0)^{\otimes k})\le c e^{-\lambda t}\frac{k}{N}W_1(\mu_0^N,\mu_0^{\otimes N})+c k\{1+\sqrt{\mu_0(|\cdot|^2)}\}N^{-1/2}$. The result relies on a careful treatment of interacting diffusion coefficients and uses auxiliary processes, gradient bounds, and integral representations to obtain uniform-in-time estimates. The findings advance the theory of propagation of chaos in singularly coupled diffusion settings and have implications for kinetic models like the Landau equation in a stochastic framework.

Abstract

In this paper, uniform in time quantitative propagation of chaos in $L^1$-Wasserstein distance for mean field interacting particle system is derived, where the diffusion coefficient is allowed to be interacting and the drift is assumed to be partially dissipative. The main tool relies on reflection coupling, the gradient estimate of the decoupled SDEs, and the Duhamel formula for two semigroups associated to two time-inhomogeneous diffusion processes on $(\R^d)^N$.

Uniform in Time Propagation of Chaos for Mean Field Particle System with Interacting Noise and Partially Dissipative Drifts

TL;DR

This work addresses uniform-in-time propagation of chaos in for a mean-field particle system with interacting diffusion coefficients and a partially dissipative drift. The authors develop a framework based on reflection coupling, gradient estimates for time-inhomogeneous decoupled SDEs, and a Duhamel formula for two semigroups to handle the coupling between the -particle system and the nonlinear limit. They prove an explicit exponential decay bound with a residual: for all and , . The result relies on a careful treatment of interacting diffusion coefficients and uses auxiliary processes, gradient bounds, and integral representations to obtain uniform-in-time estimates. The findings advance the theory of propagation of chaos in singularly coupled diffusion settings and have implications for kinetic models like the Landau equation in a stochastic framework.

Abstract

In this paper, uniform in time quantitative propagation of chaos in -Wasserstein distance for mean field interacting particle system is derived, where the diffusion coefficient is allowed to be interacting and the drift is assumed to be partially dissipative. The main tool relies on reflection coupling, the gradient estimate of the decoupled SDEs, and the Duhamel formula for two semigroups associated to two time-inhomogeneous diffusion processes on .
Paper Structure (5 sections, 6 theorems, 81 equations)

This paper contains 5 sections, 6 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['POCin']}
  4. Preparations
  5. Proof of Theorem \ref{['POCin']}

Key Result

Theorem 2.1

Assume (A) and (A') with Let $\mu_0^N\in\mathscr P_{1}((\mathbb R^d)^N)$ be exchangeable and $\mu_0\in \mathscr P_{2}(\mathbb R^d)$. Then there exist some constants $c,\lambda>0$ such that

Theorems & Definitions (10)

  • Theorem 2.1
  • Corollary 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['POCin']}