Derivative of squared W2 distance along a finite-action continuity equation

By [F1] and the AC construction [F6], choose a probability $\mathrm{\Pi }$ on $C([0,T];{\mathbb{R}}^d)$ with time marginals ${\mu }_t$ and path derivative $\dot{\gamma }(t)=v_t(\gamma (t))$ almost everywhere. Its expected action is finite. For $s<t$, Cauchy–Schwarz gives $\mathrm{\parallel }{Z}_t-Z_s{\mathrm{\parallel }}_{L^2(\mathrm{\Pi })}^2\le (t-s)\int \text{\hspace{0.17em}⁣}{\int }_s^t\mathrm{\mid }\dot{\gamma }(q){\mathrm{\mid }}^{2}dqd\mathrm{\Pi }$, so $Z_t(\gamma )=\gamma (t)$ is a continuous path in $H=L^2(\mathrm{\Pi };{\mathbb{R}}^d)$. For each partition $P=(t_i)$, the same estimate and summation give ${\displaystyle \sum _i\frac{\mathrm{\parallel }{e}_{t_{i+1}}-e_{t_i}{\mathrm{\parallel }}_{L^2(\mathrm{\Pi })}^2}{t_{i+1}-t_i}\le \int \text{\hspace{0.17em}⁣}{\int }_0^T\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}^{2}dtd\mathrm{\Pi }(\gamma )<\mathrm{\infty }\mathrm{.}}$ The path space is Polish, so [F5] makes $H=L^2(\mathrm{\Pi };{\mathbb{R}}^d)$ separable. The pathwise fundamental theorem and Fubini show $Z_t=Z_0+{\int }_0^t(\gamma \mapsto v_q(\gamma (q)))dq$ in $H$; this Bochner integral exists by the finite action. Apply [F4] to the continuous Hilbert-valued path $Z_t$: its norm derivative is the displayed Bochner integrand almost everywhere, so ${\dot{Z}}_t(\gamma )=v_t(\gamma (t))$ in $L^2(\mathrm{\Pi })$ almost everywhere.

The coupling $(Z_s,Z_t{)}_{\mathrm{\#}}\mathrm{\Pi }$ gives $W_2({\mu }_s,{\mu }_t)\le \mathrm{\parallel }{Z}_s-Z_t{\mathrm{\parallel }}_H\le {\int }_s^t\mathrm{\parallel }{\dot{Z}}_q{\mathrm{\parallel }}_{H}dq$. Hence $t\mapsto W_2({\mu }_t,\nu )$ is absolutely continuous; it is bounded on $[0,T]$, so its square is also absolutely continuous and differentiable almost everywhere. Choose a time where this scalar derivative and $Z_t$'s Hilbert derivative exist and the identities in step 1.1 hold.

For any optimal coupling ${\gamma }_t$ supplied by [F2], disintegrate $\mathrm{\Pi }$ conditionally on $Z_t=x$ using [F3] and combine that kernel with ${\gamma }_t(dx,dy)$. The resulting joint law of $(\gamma ,y)$ has path marginal $\mathrm{\Pi }$, endpoint marginal $\nu$, and $(Z_t,y)$ marginal ${\gamma }_t$. Thus $(Z_{t+h},y)$ is a coupling of ${\mu }_{t+h},\nu$, so optimality of ${\gamma }_t$ gives ${\displaystyle \frac{1}{2}{W}_2^2({\mu }_{t+h},\nu )-\frac{1}{2}{W}_2^2({\mu }_t,\nu )\le \frac{1}{2}\mathbb{E}[\mathrm{\mid }{Z}_{t+h}-y{\mathrm{\mid }}^2-\mathrm{\mid }{Z}_t-y{\mathrm{\mid }}^2]\mathrm{.}}$ Divide by $h>0$ and let $h\downarrow 0$. Differentiability of $Z$ in $L^2$ gives $\mathrm{\parallel }{Z}_{t+h}-Z_t{\mathrm{\parallel }}_H=O(h)$, so the quadratic increment term divided by $h$ tends to zero; Cauchy–Schwarz passes the linear term to $\mathbb{E}⟨Z_t-y,{\dot{Z}}_t⟩$, which equals $\int (x-y)\cdot v_t(x)d{\gamma }_t$ by the marginals and derivative identity. The left side converges to the derivative at $t$. The construction works for every optimal ${\gamma }_t$, proving the inequality.