Let $(X,d)$ be a complete separable metric space, and let $(\mathcal{P}\_2 (X), W\_2)$ be the space of probability measures on $X$ with finite second moments, equipped with the 2-Wasserstein distance. It is known that discrete measures are dense inside $(\mathcal{P}\_2 (X), W\_2)$ - namely, given any $\mu \in \mathcal{P}\_2 (X)$, and $\delta>0$, one can find a discrete measure $\mu\_\delta$ with $W\_2 (\mu, \mu\_\delta)<\delta$.


Now, let $\mu\_0, \mu\_1 \in \mathcal{P}\_2 (X)$, and let $\mu\_t$ be a $W\_2$ geodesic connecting $\mu\_0$ and $\mu\_1$ (a.k.a. $\mu\_t$ is a [not necessarily unique] displacement interpolation between $\mu\_0$ and $\mu\_1$). Is the displacement interpolation stable under discrete approximation? That is, can one pick discrete $\mu\_{0,n}, \mu\_{1,n}$ such that $\mu\_{t,n}$ is close to $\mu\_t$ for all $t\in[0,1]$?



Stability of displacement interpolation in optimal transport
The displacement interpolation $\mu\_t$ should not be fixed *a priori*, due to nonuniqueness of Wasserstein Geodesics. Thus, the correct question should be: fix the approximating sequences $(\mu\_{0,n}),(\mu\_{1,n})$ and $W\_2$ geodesics $\mu\_{t,n}$, and ask if there exists **one** $\mu\_t$ close to $\mu\_{t,n}$ for $t \in [0,1]$.


