Solving mean field rough differential equations

Abstract

We provide in this work a robust solution theory for random rough differential equations of mean field type \[ dX_{t} = V\big ( X_{t},{\mathcal L}(X_{t})\big )dt + \textrm{F} \bigl ( X_{t},{\mathcal L}(X_{t})\bigr ) dW_{t}, \] where $W$ is a random rough path and ${\mathcal L}(X_{t})$ stands for the law of $X_{t}$, with mean field interaction in both the drift and diffusivity. We show that, in addition to the enhanced path of $W$, the underlying rough path-like setting should also comprise an infinite dimensional component obtained by regarding the collection of realizations of $W$ as a deterministic trajectory with values in some $L^{q}$ space. This advocates for a suitable notion of controlled path à la Gubinelli inspired from Lions’ approach to differential calculus on Wasserstein space, the systematic use of the latter playing a fundamental role in our study. Whilst elucidating the rough set-up is a key step in the analysis, solving the mean field rough equation requires another effort: the equation cannot be dealt with as a mere rough differential equation driven by a possibly infinite dimensional rough path. Because of the mean field component, the proof of existence and uniqueness indeed asks for a specific and quite elaborated localization-in-time argument.