Dobrushin stability in 1-Wasserstein distance for mean-field flows

Fix any ${\pi }_0\in \mathrm{\Pi }({\mu }_0,{\nu }_0)$ and transport it by the two flows: ${\pi }_t=(X_t,Y_t{)}_{\mathrm{\#}}{\pi }_0\mathrm{.}$ Its marginals are ${\mu }_t,{\nu }_t$. Set $D(t)=\int \mathrm{\mid }{X}_t(x)-Y_t(y)\mathrm{\mid }d{\pi }_0(x,y)$. This is finite and continuous by the first moments, the integrable path suprema, and [F5].

For each initial pair, subtract the characteristic equations and apply [F1]. Integrating over ${\pi }_0$ yields $D(t)\le D(0)+2L{\int }_0^{t}D(s)ds,$ because the first coordinate difference contributes $LD(s)$, while the averaged interaction-coordinate difference contributes another $LD(s)$ by the marginal identities of ${\pi }_0$. Applying [L1] gives $D(t)\le e^{2Lt}D(0)$. Since ${\pi }_t$ is an admissible coupling, [F3] implies $W_1({\mu }_t,{\nu }_t)\le e^{2Lt}\int \mathrm{\mid }x-y\mathrm{\mid }d{\pi }_0$.

The inequality in step 2.1 holds for every ${\pi }_0\in \mathrm{\Pi }({\mu }_0,{\nu }_0)$. Taking the infimum of its right side gives the Statement by the definition of $W_1$ in [F3]; no minimizer is required.