Let $\mathcal{P}_2(\mathbb{R}^n)$ denote the set of Borel probability measures $\mu$ on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} |x|^2 \, d\mu(x) < \infty$. Let $\mu_0,\mu_1 \in \mathcal{P}_2(\mathbb{R}^n)$. Let $\Pi(\mu_0,\mu_1)$ denote the set of Borel probability measures $\gamma$ on $\mathbb{R}^n \times \mathbb{R}^n$ whose first marginal is $\mu_0$ and whose second marginal is $\mu_1$. Define the quadratic Wasserstein distance $W_2(\mu_0,\mu_1)$ by

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

Assume that there is a unique minimizer $\pi \in \Pi(\mu_0,\mu_1)$ of this quadratic transport cost. For each $t \in [0,1]$, define the Borel interpolation map $e_t: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ by

\begin{align*} e_t(x,y) = (1-t)x + ty. \end{align*}

Then the curve $\mu: [0,1] \to \mathcal{P}_2(\mathbb{R}^n)$ defined by

\begin{align*} \mu_t := (e_t)_\#\pi \end{align*}

is the unique constant-speed $W_2$ geodesic from $\mu_0$ to $\mu_1$ among curves obtained by Euclidean constant-speed transport of optimally paired particles. More precisely, if $\gamma \in \Pi(\mu_0,\mu_1)$ satisfies

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

and if $\nu: [0,1] \to \mathcal{P}_2(\mathbb{R}^n)$ is defined by $\nu_t := (e_t)_\#\gamma$ for $t \in [0,1]$, then $\gamma = \pi$ and $\nu_t = \mu_t$ for every $t \in [0,1]$.

In particular, if the unique optimal plan is induced by a Borel map $T: \mathbb{R}^n \to \mathbb{R}^n$, so that $\pi = (\operatorname{id},T)_\#\mu_0$, then

\begin{align*} \mu_t = ((1-t)\operatorname{id} + tT)_\#\mu_0, \quad t \in [0,1]. \end{align*}