Let $n\in\mathbb N$, and let $\mu_0,\mu_1\in\mathcal P_2(\mathbb R^n)$. Write $\Pi(\mu_0,\mu_1)$ for the set of Borel probability measures on $\mathbb R^n\times\mathbb R^n$ whose first marginal is $\mu_0$ and whose second marginal is $\mu_1$. Define

\begin{align*} W_2^2(\mu_0,\mu_1):=\inf_{\pi\in\Pi(\mu_0,\mu_1)}\int_{\mathbb R^n\times\mathbb R^n}|x-y|^2\,d\pi(x,y). \end{align*}

Then

\begin{align*} W_2^2(\mu_0,\mu_1)=\inf_{(\rho,v)}\int_0^1\int_{\mathbb R^n}|v_t(x)|^2\,d\rho_t(x)\,d\mathcal L^1(t), \end{align*}

where the infimum is taken over all pairs $(\rho,v)$ such that $\rho:[0,1]\to\mathcal P_2(\mathbb R^n)$, $t\mapsto\rho_t$, is narrowly continuous, $\rho_{t=0}=\mu_0$, $\rho_{t=1}=\mu_1$, $v:(0,1)\times\mathbb R^n\to\mathbb R^n$ is Borel, the kinetic action is finite, and the continuity equation $\partial_t\rho_t+\operatorname{div}(v_t\rho_t)=0$ holds in the distributional sense: for every $\phi\in C_c^\infty((0,1)\times\mathbb R^n)$,

\begin{align*} \int_0^1\int_{\mathbb R^n}\left(\partial_t\phi(t,x)+\nabla_x\phi(t,x)\cdot v_t(x)\right)\,d\rho_t(x)\,d\mathcal L^1(t)=0. \end{align*}

Moreover, let $m:=\mu_0\otimes\mu_1$. For every $\varepsilon>0$, define

\begin{align*} E_\varepsilon(\mu_0,\mu_1):=\inf_{\pi\in\Pi(\mu_0,\mu_1)}\left\{\int_{\mathbb R^n\times\mathbb R^n}|x-y|^2\,d\pi(x,y)+\varepsilon H(\pi\mid m)\right\}. \end{align*}

Here $H(\pi\mid m):=\int_{\mathbb R^n\times\mathbb R^n}\log(d\pi/dm)\,d\pi(x,y)$ if $\pi\ll m$, and $H(\pi\mid m):=+\infty$ otherwise. For each $\varepsilon>0$, the entropic problem defining $E_\varepsilon(\mu_0,\mu_1)$ has a unique minimizer. If there exists a sequence $(\pi_k)_{k\in\mathbb N}\subset\Pi(\mu_0,\mu_1)$ such that $\pi_k\ll m$, $H(\pi_k\mid m)<\infty$ for every $k\in\mathbb N$, and

\begin{align*} \lim_{k\to\infty}\int_{\mathbb R^n\times\mathbb R^n}|x-y|^2\,d\pi_k(x,y)=W_2^2(\mu_0,\mu_1), \end{align*}

then

\begin{align*} \lim_{\varepsilon\downarrow0}E_\varepsilon(\mu_0,\mu_1)=W_2^2(\mu_0,\mu_1). \end{align*}