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Convergence Rate for Moderate Interaction particles and Application to Mean Field Games

Josué Knorst, Christian Olivera, Alexandre B. de Souza

TL;DR

The paper develops a rigorous, rate-optimal stochastic approximation for moderately interacting particle systems with singular kernels, linking them to nonlinear Fokker–Planck equations on either 𝕋^d or ℝ^d. It introduces a Besov–Triebel–Lizorkin space–based semigroup approach to obtain trajectory-level convergence rates in both L^q and Besov-type norms, with rates determined by the moderate scaling β, the limit’s regularity (λ or η), and the spatial dimension d. The results extend to Hölder drift models and provide an explicit pathway to applying the theory to Mean Field Games by proving convergence of the MFG density under optimal feedback α=−∇u, given suitable regularity of the data. The work thus offers a quantitative, function-space–driven framework for analyzing singular-interaction particle systems and their MFG approximations, with clear implications for numerical analysis and applications in economics and physics.

Abstract

We study two interacting particle systems, both modeled as a system of $N$ stochastic differential equations driven by Brownian motions with singular kernels and moderate interaction. We show a quantitative result where the convergence rate depends on the moderate scaling parameter, the regularity of the solution of the limit equation and the dimension. Our approach is based on the techniques of stochastic calculus, some properties of Besov and Triebel-Lizorkin space, and the semigroup approach introduced in [9].

Convergence Rate for Moderate Interaction particles and Application to Mean Field Games

TL;DR

The paper develops a rigorous, rate-optimal stochastic approximation for moderately interacting particle systems with singular kernels, linking them to nonlinear Fokker–Planck equations on either 𝕋^d or ℝ^d. It introduces a Besov–Triebel–Lizorkin space–based semigroup approach to obtain trajectory-level convergence rates in both L^q and Besov-type norms, with rates determined by the moderate scaling β, the limit’s regularity (λ or η), and the spatial dimension d. The results extend to Hölder drift models and provide an explicit pathway to applying the theory to Mean Field Games by proving convergence of the MFG density under optimal feedback α=−∇u, given suitable regularity of the data. The work thus offers a quantitative, function-space–driven framework for analyzing singular-interaction particle systems and their MFG approximations, with clear implications for numerical analysis and applications in economics and physics.

Abstract

We study two interacting particle systems, both modeled as a system of stochastic differential equations driven by Brownian motions with singular kernels and moderate interaction. We show a quantitative result where the convergence rate depends on the moderate scaling parameter, the regularity of the solution of the limit equation and the dimension. Our approach is based on the techniques of stochastic calculus, some properties of Besov and Triebel-Lizorkin space, and the semigroup approach introduced in [9].
Paper Structure (13 sections, 7 theorems, 113 equations)

This paper contains 13 sections, 7 theorems, 113 equations.

Table of Contents

  1. Introduction and results.
  2. Fokker-Planck equations with singular kernel.
  3. Fokker-Planck equations with Hölder drift term.
  4. Notations.
  5. Bessel spaces.
  6. Triebel-Lizorkin and Besov Space.
  7. First main result.
  8. Second main result.
  9. Application to Mean Field Games
  10. Proof of Theorem \ref{['th:rate2']}
  11. Proof of Theorem \ref{['mainthm']}
  12. Appendix
  13. Appendix

Key Result

Theorem 1

Assume that $K \in L^{q^\prime}(\mathbb{T}^d)$ and there exists $C_K >0$ such that for any $f\in L^{q}(\mathbb{T}^{d})$ with $q>d/\lambda$ and $\lambda >0$. Let $T_{\max}$ be the maximal existence time for eq:PDE, fix $T\in(0,T_{\max})$ and take $\delta>0$ such that $\rho >0$, where In addition, let the dynamics of the particle system be given by itoass and assume and with $q'$ and $r'$ as the

Theorems & Definitions (12)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1
  • Definition 1.6
  • Theorem 2
  • Proposition 1.8
  • proof
  • Proposition 1.9
  • Proposition A.1
  • proof
  • ...and 2 more