Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp propagation of chaos for mean field Langevin dynamics, control, and games

Manuel Arnese, Daniel Lacker

Abstract

We establish the sharp rate of propagation of chaos for McKean-Vlasov equations with coefficients that are non-linear in the measure argument, i.e., not necessarily given by pairwise interactions. Results are given both on bounded time horizon and uniform in time. As applications, we deduce the sharp rate of propagation of chaos for the convergence problem in mean field games and control, and for mean field Langevin dynamics, the latter being uniform in time in the strongly displacement convex regime. Our arguments combine the BBGKY hierarchy with techniques from the literature on weak propagation of chaos.

Sharp propagation of chaos for mean field Langevin dynamics, control, and games

Abstract

We establish the sharp rate of propagation of chaos for McKean-Vlasov equations with coefficients that are non-linear in the measure argument, i.e., not necessarily given by pairwise interactions. Results are given both on bounded time horizon and uniform in time. As applications, we deduce the sharp rate of propagation of chaos for the convergence problem in mean field games and control, and for mean field Langevin dynamics, the latter being uniform in time in the strongly displacement convex regime. Our arguments combine the BBGKY hierarchy with techniques from the literature on weak propagation of chaos.
Paper Structure (21 sections, 29 theorems, 217 equations)

This paper contains 21 sections, 29 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Analysis on the space of measures
  4. Assumptions and main results
  5. Comments on the assumptions
  6. Application: Mean Field Langevin Dynamics
  7. Application: Mean Field Games
  8. Application: Mean Field Control
  9. Related literature
  10. Outline
  11. Entropy inequality via BBGKY hierarchy
  12. Hierarchies of linear differential inequalities
  13. Short time Analysis
  14. Uniform in Time Analysis
  15. Flows and semigroups over $\mathcal{P}_2({\mathbb R}^d)$
  16. ...and 6 more sections

Key Result

Theorem 2.3

Under assumption assumption on Phi, for every $k\leq n$ and every fixed $t>0$, we have The hidden constant depends on $t$, $C_{\textup{T}_1}(0)$, $\mu_0$, and the derivative bounds on $V$.

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8
  • Corollary 2.9
  • Example 2.10
  • ...and 50 more