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!
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):