Generalized W2 geodesics and strong Wasserstein subgradients

Assume AC. Let ${\mu }_0,{\mu }_1\in {\mathcal{P}}_2({\mathbb{R}}^d)$. A generalized $W_2$ geodesic from ${\mu }_0$ to ${\mu }_1$ with base $\nu \in {\mathcal{P}}_2({\mathbb{R}}^d)$ is a probability measure $\pi$ on triples $(z,x,y)$ whose $(z,x)$ marginal is an optimal coupling of $\nu$ and ${\mu }_0$ and whose $(z,y)$ marginal is an optimal coupling of $\nu$ and ${\mu }_1$; its interpolating curve is ${\mu }_t=((1-t)x+ty{)}_{\mathrm{\#}}\pi$. A functional $F$ is $\lambda$-convex along generalized geodesics if for every such $\pi$ its values satisfy $F({\mu }_t)\le (1-t)F({\mu }_0)+tF({\mu }_1)-\frac{\lambda }{2}t(1-t)\int \mathrm{\mid }x-y{\mathrm{\mid }}^{2}d\pi$. A vector field $g_{\mu }\in L^2(\mu ;{\mathbb{R}}^d)$ is a strong Wasserstein subgradient of $F$ at $\mu$ if for every $\eta \in {\mathcal{P}}_2({\mathbb{R}}^d)$ and every coupling $\gamma$ of $\mu$ and $\eta$, $F(\eta )\ge F(\mu )+\int g_{\mu }(x)\cdot (y-x)d\gamma +o\text{\hspace{0.17em}⁣}((\int \mathrm{\mid }y-x{\mathrm{\mid }}^{2}d\gamma {)}^{1\mathrm{/}2})$ as the coupling action tends to zero, with a remainder uniform over all such couplings. The definition imposes no global displacement-convexity claim.

The Euclidean quadratic Wasserstein space

Optimal quadratic couplings exist and their cost is weakly lower semicontinuous

Ambrosio, Gigli, and Savaré, Definitions 9.2.4 and 10.3.1

Chizat and Bach, Appendix B.2.1