Propagation of chaos for mean field rough differential equations

Abstract

We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type

$$\mathit{d}{\mathit{X}}_{\mathit{t}}=\text{V}({\mathit{X}}_{\mathit{t}},\mathcal{L}({\mathit{X}}_{\mathit{t}}))\mathit{d}\mathit{t}+\text{F}({\mathit{X}}_{\mathit{t}},\mathcal{L}({\mathit{X}}_{\mathit{t}}))\mathit{d}{\mathit{W}}_{\mathit{t}},$$

where W is a random rough path and $\mathcal{L}({\mathit{X}}_{\mathit{t}})$ is the law of ${\mathit{X}}_{\mathit{t}}$. 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 (Electron. J. Probab. 25 (2020) 21) 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.