Let $(M,g)$ be a complete connected smooth Riemannian manifold without boundary. Let $d_g:M\times M\to[0,\infty)$ denote its Riemannian distance, let $\operatorname{vol}_g$ denote its Riemannian volume measure on $(M,\mathcal B(M))$, and let $V\in C^\infty(M;\mathbb R)$. Define the weighted Borel measure $m$ on $(M,\mathcal B(M))$ by $dm=e^{-V}\,d\operatorname{vol}_g$. Assume that $m$ is locally finite and has full support.

Let $\mathcal P_2(M)$ denote the set of Borel probability measures $\mu$ on $M$ satisfying

\begin{align*} \int_M d_g(x,x_0)^2\,d\mu(x)<\infty \end{align*}

for some, equivalently every, $x_0\in M$, and let $W_2$ denote the quadratic Wasserstein distance on $\mathcal P_2(M)$ induced by $d_g$.

Let $K\in\mathbb R$, and assume that the Bakry-Emery Ricci tensor $\operatorname{Ric}_V:=\operatorname{Ric}+\operatorname{Hess}_g V$ satisfies

\begin{align*} \operatorname{Ric}_V|_x(\xi,\xi)\ge K g_x(\xi,\xi) \end{align*}

for every $x\in M$ and every $\xi\in T_xM$.

For $\mu\in\mathcal P_2(M)$, define the relative entropy of $\mu$ with respect to $m$ by

\begin{align*} \operatorname{Ent}_m(\mu):=\int_M \rho\log\rho\,dm \end{align*}

if $\mu=\rho m$ for some Borel function $\rho:M\to[0,\infty)$ and the integral is well-defined in $(-\infty,+\infty]$, and set $\operatorname{Ent}_m(\mu)=+\infty$ otherwise.

Then for every $\mu_0,\mu_1\in\mathcal P_2(M)$ satisfying $-\infty<\operatorname{Ent}_m(\mu_0)<+\infty$ and $-\infty<\operatorname{Ent}_m(\mu_1)<+\infty$, there exists a constant-speed $W_2$-geodesic $\mu:[0,1]\to\mathcal P_2(M)$, $t\mapsto\mu_t$, from $\mu_0$ to $\mu_1$ such that, for every $t\in[0,1]$,

\begin{align*} \operatorname{Ent}_m(\mu_t)\le (1-t)\operatorname{Ent}_m(\mu_0)+t\operatorname{Ent}_m(\mu_1)-\frac{K}{2}t(1-t)W_2^2(\mu_0,\mu_1) \end{align*}

Equivalently, for every pair of finite-entropy endpoints in $\mathcal P_2(M)$, there is at least one optimal dynamical plan whose time marginals satisfy the above $K$-convexity inequality; in this sense $\operatorname{Ent}_m$ is $K$-displacement convex on its finite-entropy domain in the weak optimal-plan sense.