# Stability of displacement interpolation in optimal transport

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 $$t\in[0,1]$$?

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 $$t \in [0,1]$$.
• This is a comment rather than an answer, but I could not post it as a comment. Anyway, something useful in this direction can be found in Lemma 4.4 arxiv.org/pdf/1609.00782.pdf which, combined with Proposition 4.8 of arxiv.org/pdf/1311.4907.pdf gives you $W_2$ close $\mu_{t,n}$'s. Commented Aug 19, 2020 at 8:10
• Certainly there are non-uniqueness issues. I actually meant something like: given a Wasserstein geodesic $\mu_t$, can we produce sequences $(\mu_{0,n})$ and $(\mu_{1,n})$ such that $(\mu_{t,n})$ converges to $\mu_t$ in some suitable sense. Commented Aug 19, 2020 at 8:35
• I think is very hard to fix $\mu_t$ and produce, afterwards, approximating marginals $(\mu_{0,n}),(\mu_{1,n})$. If you stick to the other way, as in my answer, you can try to argue essentially by tightness to get $\mu_t$ from $\mu_{t,n}$. Finally, you just need to show that the limit $\mu_t$ is a $W_2$ geodesic. The first link I gave you follows this path. With this approach, is very hard to control whether you are converging to your fixed a priori Wasserstein geodesic, or to another. Commented Aug 19, 2020 at 8:43
• There is not only the problem of uniqueness, but also of existence. Under the present assumptions, $\mu_{0,n}$ and $\mu_{1,n}$ may not be connected by a $W_2$-geodesic. Commented Aug 19, 2020 at 9:21