Let $d\in\mathbb N$. Let $\mu_0,\mu_1\in\mathcal P(\mathbb R^d)$ be compactly supported Borel probability measures such that $\mu_0\ll\mathcal L^d$ and $\mu_1\ll\mathcal L^d$. Write $d\mu_1=\rho_1\,d\mathcal L^d$, and assume

\begin{align*} \int_{\mathbb R^d}\rho_1(y)\log \rho_1(y)\,d\mathcal L^d(y)<\infty. \end{align*}

For Borel probability measures $\alpha$ and $\beta$ on $\mathbb R^d$, let $\Pi(\alpha,\beta)$ denote the set of Borel probability measures on $\mathbb R^d\times\mathbb R^d$ whose first marginal is $\alpha$ and whose second marginal is $\beta$. For each $\varepsilon>0$, let $R_{\mu_0}^{\varepsilon}$ denote the Borel probability measure on $C([0,1];\mathbb R^d)$ induced by the process

\begin{align*} X_t=X_0+\sqrt{\varepsilon}B_t,\qquad t\in[0,1], \end{align*}

where $X_0$ has law $\mu_0$ and $B:[0,1]\to\mathbb R^d$ is a standard Brownian motion independent of $X_0$. Let $X_t:C([0,1];\mathbb R^d)\to\mathbb R^d$ denote the evaluation map $X_t(\omega)=\omega(t)$. For each $\varepsilon>0$, let $P^\varepsilon$ be a minimizer of

\begin{align*} \inf\left\{H(P\mid R_{\mu_0}^{\varepsilon}):P\in\mathcal P(C([0,1];\mathbb R^d)),\ (X_0)_\#P=\mu_0,\ (X_1)_\#P=\mu_1\right\}, \end{align*}

where $H(\cdot\mid\cdot)$ denotes relative entropy. For each $\varepsilon>0$, set

\begin{align*} \pi^\varepsilon:=(X_0,X_1)_\#P^\varepsilon. \end{align*}

If $(\varepsilon_n)_{n=1}^{\infty}\subset(0,\infty)$ satisfies $\varepsilon_n\downarrow0$ and $\pi^{\varepsilon_n}$ converges weakly to a Borel probability measure $\pi$ on $\mathbb R^d\times\mathbb R^d$, then $\pi\in\Pi(\mu_0,\mu_1)$ and

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

Thus $\pi$ is an optimal coupling of $\mu_0$ and $\mu_1$ for the quadratic cost $(x,y)\mapsto |x-y|^2/2$.