Finite-action continuity equations generate AC2 Wasserstein curves

For $0\le s<t\le T$, the evaluation map $E_{s,t}:C([0,T];{\mathbb{R}}^d)\to {\mathbb{R}}^{2d}$, $E_{s,t}(\gamma )=(\gamma (s),\gamma (t))$, is continuous. Define ${\pi }_{s,t}(B)=\eta (E_{s,t}^{-1}(B))$ for Borel $B\subseteq {\mathbb{R}}^{2d}$; by [F4] this is a Borel probability. Its coordinate marginals are $(e_s{)}_{\mathrm{\#}}\eta ={\mu }_s$ and $(e_t{)}_{\mathrm{\#}}\eta ={\mu }_t$, so ${\pi }_{s,t}$ is a coupling. Its quadratic cost satisfies $\int \mathrm{\mid }x-y{\mathrm{\mid }}^{2}d{\pi }_{s,t}(x,y)=\int \mathrm{\mid }\gamma (t)-\gamma (s){\mathrm{\mid }}^{2}d\eta (\gamma )$ by [F4]. For $\eta$-almost every path, absolute continuity gives $\mathrm{\mid }\gamma (t)-\gamma (s)\mathrm{\mid }\le {\int }_s^t\mathrm{\mid }\dot{\gamma }(r)\mathrm{\mid }dr$. Minkowski [F2], applied with the path probability space and Lebesgue measure on $[s,t]$, yields $W_2({\mu }_s,{\mu }_t)\le {(\int {({\int }_s^t\mathrm{\mid }\dot{\gamma }(r)\mathrm{\mid }dr)}^{2}d\eta )}^{1\mathrm{/}2}\le {\int }_s^t{(\int \mathrm{\mid }\dot{\gamma }(r){\mathrm{\mid }}^{2}d\eta )}^{1\mathrm{/}2}dr\mathrm{.}$ The marginal identity and the pathwise velocity identity in [F1] identify the inner integral for almost every r as $\int \mathrm{\mid }{v}_r(x){\mathrm{\mid }}^{2}d{\mu }_r(x)$. Put $g(r)=(\int \mathrm{\mid }{v}_r{\mathrm{\mid }}^{2}d{\mu }_r{)}^{1\mathrm{/}2}$. Finite action gives $g\in L^2(0,T)$, and the displayed bound for every interval implies W2-continuity and the defining $A{C}^2$ upper-control inequality.

By [F3], the metric derivative exists almost everywhere and is no larger than any nonnegative $L^2$ upper control. Applying it to $g$ from step 1.1 gives $\mathrm{\mid }{\dot{\mu }}_t\mathrm{\mid }\le g(t)$ almost everywhere, as claimed.