Exact quadratic feature-energy flow and particle limit

The characteristic equation $\dot{x}=-x$ with $x(0)=x_0$ has the solution $x(t)=e^{-t}{x}_0$, as direct differentiation verifies. Pushing ${\mu }_0$ forward by this map gives a continuity-equation solution with velocity $v_t(x)=-x$. Its action on $[0,T]$ is finite because $\int \mathrm{\mid }x{\mathrm{\mid }}^{2}d{\mu }_t=e^{-2t}\int \mathrm{\mid }x{\mathrm{\mid }}^{2}d{\mu }_0$. For compactly supported initial laws it is the flow identified by the global uniqueness theorem. For arbitrary ${\mu }_0\in {\mathcal{P}}_2$, the same conclusion follows directly: every finite-action PDE solution with the prescribed smooth velocity $-x$ is represented by its unique characteristics via superposition, so it equals this pushforward, even when ${\mu }_0$ lies in no finite cutoff.

The metric $m^{-1}{\sum }_i{x}_i\cdot y_i$ gives the particle gradient of $m^{-1}{\sum }_i\mathrm{\mid }{x}_i{\mathrm{\mid }}^2\mathrm{/}2$ equal to $(x_1,\dots ,x_m)$, so the same scalar ODE gives $x_i(t)=e^{-t}{x}_i(0)$. Consequently ${\mu }_{m,t}=(e^{-t}\mathrm{id}{)}_{\mathrm{\#}}{\mu }_{m,0}$.

For $a>0$, pushing both marginals of any coupling forward by $x\mapsto ax$ multiplies its quadratic cost by $a^2$; applying the inverse scaling gives the reverse inequality for the infimum. Thus $W_2(a_{\mathrm{\#}}\alpha ,a_{\mathrm{\#}}\beta )=a{W}_2(\alpha ,\beta )$. Apply this with $a=e^{-t}$, $\alpha ={\mu }_{m,0}$, and $\beta ={\mu }_0$ to obtain the displayed identity. Since $e^{-t}\le 1$, its supremum for $0\le t\le T$ is at most the initial W2 error, proving compact-time uniform convergence.