Assume that $\rho$ and all its derivatives are rapidly decreasing. Let $\mathcal F$ be a real-valued functional on a class of such densities. Suppose that $\mathcal F$ has a first variation at $\rho$ in the following sense: there exists $u_\rho\in C^\infty(\mathbb R^n)$, unique up to an additive constant, whose derivatives have at most polynomial growth, such that for every $\eta\in C_c^\infty(\mathbb R^n)$ satisfying
Equip the smooth Wasserstein tangent space at $\rho$ with the Otto convention that a gradient vector field $v=\nabla\phi$, with $\phi\in C_c^\infty(\mathbb R^n)$, represents the density variation $-\nabla\cdot(\rho v)$ and has inner product
where the additive constant ambiguity in the first variation does not affect the right-hand side.
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Consequently, if $I\subset\mathbb R$ is an interval, if $(\rho_t)_{t\in I}$ is a smooth curve of densities satisfying the same regularity assumptions, and if