Finite-particle flows converge to the measure-valued flow in W2

Put the finite labels on the probability space $\{1,\dots ,m\}$ with mass $1\mathrm{/}m$. The finite label sigma-algebra is countably generated, so the generalized label lemma gives a separable Hilbert space. Then $K=L^2(\{1,\dots ,m\};{\mathbb{R}}^d)$ has inner product $m^{-1}{\sum }_i{x}_i\cdot y_i$. For each cutoff $r$, its localized energy $E_r$ equals $F(m^{-1}{\sum }_i{\delta }_{x_i})=F_m$ on $Q_r^m$ and is $+\mathrm{\infty }$ outside that set by [F1]. A Euclidean subgradient $p$ represents the functional derivative in the weighted metric by $mp$, since $m^{-1}{\sum }_i(m{p}_i)\cdot h_i={\sum }_i{p}_i\cdot h_i$. This gives the normalization of the two inclusions; the local supporting-set argument below proves their local subdifferential identification and the converse for every particle path.

Fix $m$, a finite interval $[0,T]$, and any classical particle path from the Statement. Its coordinate image is compact. Fix $q\in Q_1$ and first choose $r>1+{\max }_{i,t}\mathrm{\mid }{x}_i^{(m)}(t)-q\mathrm{\mid }$. The cutoff enlargement condition then puts the image in $Q_r^m$. Choose $R>r$; the same condition gives a relative ${\mathrm{\Omega }}^m$-neighborhood of the image inside $Q_R^m$. Set $A_R={\sup }_{Q_R}\mathrm{\parallel }d\mathrm{\Phi }\mathrm{\parallel }$, $B_R={\mathrm{Lip}}_{Q_R}(d\mathrm{\Phi })$, and choose the finite Lipschitz and norm bounds $L_R,N_R$ for $dR$ on the closed convex hull of $\mathrm{\Phi }(Q_R)$. On the atomic label space write $G_R(X)=R(\int \mathrm{\Phi }(X)d\sigma )$. Its gradient in the weighted Hilbert metric is $s\mapsto d{\mathrm{\Phi }}_{X(s)}^{\ast }\mathrm{\nabla }R(\int \mathrm{\Phi }(X)d\sigma )$. Also $\mathrm{\parallel }\int \mathrm{\Phi }(X)d\sigma -\int \mathrm{\Phi }(Y)d\sigma \mathrm{\parallel }\le A_R\mathrm{\parallel }X-Y{\mathrm{\parallel }}_K$. Adding and subtracting $d{\mathrm{\Phi }}_{X(s)}^{\ast }\mathrm{\nabla }R(\int \mathrm{\Phi }(Y)d\sigma )$ therefore gives the Lipschitz bound $\mathrm{\parallel }\mathrm{\nabla }{G}_R(X)-\mathrm{\nabla }{G}_R(Y){\mathrm{\parallel }}_K\le (L_R{A}_R^2+N_R{B}_R)\mathrm{\parallel }X-Y{\mathrm{\parallel }}_K\mathrm{.}$ Writing $C_R=L_R{A}_R^2+N_R{B}_R$, the smooth term is therefore $(-C_R)$-semiconvex in the weighted Hilbert metric, while the averaged potential is ${\lambda }_V$-semiconvex there. Since $\mathrm{\parallel }X-Y{\mathrm{\parallel }}_K^2=m^{-1}{\sum }_i\mathrm{\mid }{x}_i-y_i{\mathrm{\mid }}^2$, the corresponding Euclidean product coefficient for $F_m$ is $({\lambda }_V-C_R)\mathrm{/}m$. Because $Q_R^m$ and ${\mathrm{\Omega }}^m$ agree locally along the path, the local Fréchet subdifferential in the Statement is the supporting subdifferential on $Q_R^m$ with this coefficient. Under the weighted Hilbert norm, $p\in \mathrm{\partial }{F}_m(x)$ corresponds exactly to $mp\in {\mathrm{\partial }}_{{\lambda }_V-C_R}{E}_R(x)$.

On a bounded neighborhood of the compact path image, the smooth feature gradient is bounded, and the two supporting inequalities for $V$ together with its bounded supporting selections give a local Lipschitz bound for the potential. Hence $F_m$ is locally Lipschitz there, so $e(t)=F_m(x^{(m)}(t))$ is absolutely continuous. At almost every time when $e$ and $x^{(m)}$ are differentiable and the inclusion holds, apply the semiconvex supporting inequality at $x^{(m)}(t)$ to $x^{(m)}(t+h)$ for positive and negative $h$. Dividing by $h$ and letting $h\to 0$ gives $e^{\mathrm{\prime }}(t)={\sum }_i{p}_i(t)\cdot {\dot{x}}_i^{(m)}(t)=-m{\sum }_i\mathrm{\mid }{p}_i(t){\mathrm{\mid }}^2=-\mathrm{\parallel }{\dot{x}}^{(m)}(t){\mathrm{\parallel }}_K^2\mathrm{.}$ Since $F_m\ge 0$, integration shows that the path has square-integrable derivative on $[0,T]$. Step 2.1 therefore identifies it as a solution of the localized Hilbert gradient-flow inclusion, and uniqueness of that flow identifies it with the atomic localized flow. The escape estimate [F3] consequently applies to every particle path in the Statement.

By [F3], for each fixed $T$ the particle flows, now identified with localized flows, and the measure flow remain inside a common larger cutoff $Q_R$ and solve the same localized equation. Apply [F4] on that cutoff to get $W_2({\mu }_{m,t},{\mu }_t)\le e^{K_{T}t}{W}_2({\mu }_{m,0},{\mu }_0)$ for $0\le t\le T$. Since $e^{K_{T}t}\le e^{K_{T}T}$ when $K_T\ge 0$ (replace it by $\max (K_T,0)$ otherwise), taking the supremum and using the assumed initial convergence proves the result.