The first Fundamental Theorem of Calculus for functions defined on Wasserstein space

Abstract

We establish an analogue of the first fundamental theorem of calculus for functions defined on the Wasserstein space of probability measures. Precisely, we show that if a function on the Wasserstein space is sufficiently regular in the sense of the linear functional derivative, then its integral is differentiable and the derivative coincides with the integrand. Our approach relies on a general differentiability criterion that connects the linear functional derivative, viewed as a Fr\'echet-derivative, and Dawson's weaker notion, which corresponds to a Gateaux-derivative. Under suitable regularity assumptions, it is possible to upgrade Gateaux-differentiability to Fr\'echet-differentiability in the infinite-dimensional setting of Wasserstein space.