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Let $\mathcal{P}_2$ the space of absolutely continuous probability measures on $\mathbb{R}^d$ with finite second moment equipped with the $2$-Wasserstein metric. Fix $\mu_0, \mu_1 \in \mathcal{P}_2.$ Let $T$ be the optimal transport map from $\mu_0$ to $\mu_1.$ It is well known that the geodesic curve $\mu_t = ((1 - t) \text{id} + t T)_\# \mu_0$ is absolutely continuous.

I am wondering if the linear interpolation $\tilde \mu_t = (1 - t) \mu_0 + t \mu_1$ is absolutely continuous.

Any reference would be helpful too.

Thank you!

Paruru
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1 Answers1

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They are not. The reason is basically given in this post.

As a simple counterexample, consider $\mu_0=\delta_0$ and $\mu_1=\delta_1$. Then $W_2(\mu_0,\mu_t)=\sqrt t$ and hence $$\lim_{t\to 0}\frac{W_2(\mu_0,\mu_t)}{t}=\infty.$$

Hence $(\mu_t)_{t\in[0,1]}$ cannot be absolutely continuous. Here we have used the following result (see Lectures on Optimal Transport):

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Small Deviation
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  • Thank you! How about under the assumption that $\mu_0$ and $\mu_1$ have densities? – Paruru Oct 08 '23 at 10:52
  • OK, having densities does not matter. There is still an example where this happens. – Paruru Oct 08 '23 at 22:31
  • Are you aware of any example where $\tilde \mu_t$ is not absolutely continuous even though $\tilde \mu_t$ has a common support? – Paruru Oct 08 '23 at 22:59
  • @Paruru What if we pick some measure $\mu\in\mathcal P_2$ such that the support of $\mu$ contains $0$ and $1$; then put $\mu_0=\frac 12(\delta_0+\mu)$ and $\mu_1=\frac 12(\delta_1+\mu)$? – Alex Ravsky Oct 17 '23 at 17:17