[Paper Review] Propagation of chaos for mean field rough differential equations

Ismaël Bailleul, Rémi Catellier+1 more|arXiv (Cornell University)|Jul 1, 2019

Stochastic processes and statistical mechanics30 references4 citations

TL;DR

This paper establishes propagation of chaos for mean field rough differential equations driven by random rough paths, proving that the empirical distribution of a large system of interacting particles converges to the solution of a nonlinear mean field SDE. Using a rough path framework and a coupling argument inspired by Sznitman, the authors derive an explicit optimal convergence rate of order $ n^{-1/2} $, extending classical results beyond Itô calculus to rough differential equations with mean field interaction in the diffusion coefficient.

ABSTRACT

We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ is the law of $X_t$. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper [1] for solving mean field rough differential equations and in particular upon a corresponding version of the Itô-Lyons continuity theorem. The rate of convergence is obtained by a coupling argument developed first by Sznitman for particle systems with Brownian inputs.

Motivation & Objective

• To establish propagation of chaos for mean field rough differential equations where the diffusion coefficient depends on the law of the solution, extending beyond classical Itô calculus.
• To provide a rigorous convergence rate for the empirical measure of $ n $-particle systems to the solution of the corresponding mean field equation.
• To extend the applicability of rough path theory to mean field models with nonlinear interaction in the diffusion coefficient.
• To develop a coupling-based argument to derive optimal convergence rates, generalizing Sznitman's method to the rough path setting.

Proposed method

• The analysis is based on a version of the Itô-Lyons continuity theorem adapted to mean field rough differential equations, ensuring pathwise uniqueness and stability.
• The authors use Lions' calculus on Wasserstein space to handle the nonlinear dependence of the diffusion coefficient on the law of the solution.
• A coupling argument is employed to compare the $ n $-particle system with the mean field limit, leveraging moment bounds and the $ p $-variation of rough paths.
• The solution is constructed via a fixed-point argument in a space of rough paths with controlled weighted norms, ensuring convergence in the $ \mathbb{L}^\varrho $-norm.
• The method relies on enhanced rough paths, including iterated integrals of the driving signals, to define the rough differential equation in a pathwise manner.
• The convergence rate is derived by estimating the difference between the empirical measure and the law of the solution using moment estimates and the Wasserstein distance.

Experimental results

Research questions

• RQ1 Does the empirical measure of a system of $ n $ interacting rough differential equations converge to the solution of a mean field rough SDE as $ n \to \infty $?
• RQ2 What is the optimal rate of convergence for such propagation of chaos in the rough path setting?
• RQ3 Can Sznitman's coupling method be adapted to derive convergence rates in the context of rough differential equations with mean field interaction?
• RQ4 How does the dependence of the diffusion coefficient on the law of the solution affect the well-posedness and propagation of chaos?
• RQ5 Under what regularity assumptions on the coefficients and the driving rough path does propagation of chaos hold?

Key findings

• Propagation of chaos holds for mean field rough differential equations with general nonlinear dependence of the diffusion coefficient on the law of the solution.
• The convergence rate of the $ n $-particle system to the mean field limit is proven to be optimal at order $ n^{-1/2} $.
• The convergence is established in the $ \mathbb{L}^\varrho $-norm for any $ \varrho \geq 1 $, with explicit moment bounds.
• The proof relies on a coupling argument that extends Sznitman's method to the rough path setting, enabling precise rate estimation.
• The analysis is based on a rough path version of the Itô-Lyons theorem, ensuring stability and pathwise uniqueness for the mean field equation.
• The authors derive uniform moment estimates for the solution and its empirical approximation, which are crucial for the convergence rate analysis.