Let $n\in\mathbb N$, let $I\subset\mathbb R$ be an open interval with $0\in I$, and let $V\in C^2(\mathbb R^n;\mathbb R)$. Define the potential energy functional $\mathcal V$ on densities $\rho$ for which the integral is finite by

\begin{align*} \mathcal V[\rho]=\int_{\mathbb R^n}V(x)\rho(x)\,d\mathcal L^n(x). \end{align*}

Let $\rho:I\times\mathbb R^n\to[0,\infty)$, $(t,x)\mapsto\rho_t(x)$, be a smooth curve of probability densities, and let $\phi:I\times\mathbb R^n\to\mathbb R$, $(t,x)\mapsto\phi_t(x)$, be smooth. Assume that there exist an open interval $I_0\subset I$ with $0\in I_0$ and a compact set $K\subset\mathbb R^n$ such that $\operatorname{supp}\rho_t\subset K$ for every $t\in I_0$. Assume also that, for every $t\in I_0$,

\begin{align*} \int_{\mathbb R^n}\rho_t(x)\,d\mathcal L^n(x)=1, \end{align*}

and that $(\rho_t)_{t\in I_0}$ is a smooth formal $W_2$-geodesic with velocity field $\nabla\phi_t$, meaning that

\begin{align*} \partial_t\rho_t+\nabla\cdot(\rho_t\nabla\phi_t)=0 \end{align*}

on $I_0\times\mathbb R^n$, and that there is a smooth function $c:I_0\to\mathbb R$ such that

\begin{align*} \partial_t\phi_t+\frac{1}{2}|\nabla\phi_t|^2=c(t) \end{align*}

on $I_0\times\mathbb R^n$. Then the formal Wasserstein Hessian of $\mathcal V$ at $\rho_0$ in the tangent direction $\nabla\phi_0$ is

\begin{align*} \operatorname{Hess}_{W_2}\mathcal V(\rho_0)[\nabla\phi_0,\nabla\phi_0]=\left.\frac{d^2}{dt^2}\right|_{t=0}\mathcal V[\rho_t]=\int_{\mathbb R^n}\nabla\phi_0(x)\cdot\bigl(D^2V(x)\nabla\phi_0(x)\bigr)\rho_0(x)\,d\mathcal L^n(x). \end{align*}

Consequently, if $D^2V(x)$ is positive semidefinite for every $x\in\mathbb R^n$, then $\mathcal V$ is formally displacement convex along the smooth compactly supported Wasserstein geodesics in this class.