Solving mean field rough differential equations
The authors consider the equations of MacKean-Vlasov type with rough path, or, in other words, random rough differential equations of mean field type. The precise purpose of the paper is to study the equation of the form \( dX_t(\omega)= F(X_t(\omega), \mathcal{L}(X_t))dW_t(\omega)\), where \(\mathcal{L}(A)\) is the law of the random variable \(A\), and the raw driver \(W_t(\omega)\) is assumed to take values in some \(\mathbb{R}^m\) and to be \(1/p\)-Hölder continuous, for \(p\in[2, 3)\). \textit{M. Gubinelli}'s versatile approach [J. Funct. Anal. 216, No. 1, 86--140 (2004; Zbl 1058.60037)] of controlled paths is used to make sense of this equation. It is shown that, in addition to the enhanced path of \(W\), the underlying rough path-like setting also comprises 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 the paper. Because of the mean field component, the proof of existence and uniqueness asks for a specific and quite elaborated localization-in-time argument. The authors formulate and carefully investigate regularity conditions on \(F\), and then prove the main existence-uniqueness theorem, global in the sense that it exists on the whole prescribed interval.
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