An Elementary Proof for the Structure of Wasserstein Derivatives

Let $F:{\mathbb{L}}^2(\mathrm{\Omega },\mathbb{R})\to \mathbb{R}$ be a law invariant and continuously Fr'echet differentiable mapping. Based on Lions \cite{Lions}, Cardaliaguet \cite{Cardaliaguet} (Theorem 6.2 and 6.5) proved that: \bea \label{Derivative} D F (\xi) = g(\xi), \eea where $g: $\mathbb{R}$ \to $\mathbb{R}$$isadeterministicfunctionwhichdependsonlyonthelawof$\xi$. See also Carmona & Delarue \cite{CD} Section 5.2. In this short note we provide an elementary proof for this well known result. This note is part of our accompanying paper \cite{WZ}, which deals with a more general situation.