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.
Highlights
The first works on mean field stochastic dynamics and interacting diffusions/Markov processes have their roots in Kac’s simplified approach to kinetic theory [28] and McKean’s work [34] on nonlinear parabolic equations
(where LpAq stands for the law of a random variable A) and relate it to the empirical behaviour of large systems of interacting dynamics
The main emphasis of subsequent works has been on proving propagation of chaos and other limit theorems, and giving stochastic representations of solutions to nonlinear parabolic equations under more and more general settings; see [36, 37, 25, 17, 18, 35, 27, 7, 8] for a tiny sample
Summary
The first works on mean field stochastic dynamics and interacting diffusions/Markov processes have their roots in Kac’s simplified approach to kinetic theory [28] and McKean’s work [34] on nonlinear parabolic equations. The solutions are constructed up to a random time, say ζ, yielding a random path pxtq0ďtďζ defined up to ζ, but, for such solutions, we can only make sense of Lpxt^ζ q rather than Lpxtq, for t ě 0 This is a serious drawback for solving mean field rough equations, unless we know a priori that ζ is infinite, as is the case in Cass and Lyons’ work. The striking fact of the analysis performed in [4] is based upon an observation noticed first by Tanaka in his seminal work [38] on limit theorems for mean field type diffusions, and used crucially by Cass and Lyons in [13] It says that, for a given ω P Ω, the aforementioned particle system associated with (1.2) may be interpreted as a mean field rough equation (in the sense of our Definition 4.1 below) but with respect to the empirical version of the rough setting. As for processes X‚ “ pXtqtPI , defined on a time interval I, we often write X for X‚
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