Proof of A Configuration-Level Majorant of the Distance from a One-Particle Marginal, Touching at a Given Configuration Law

Each result cited is universally quantified over the data in its own statement. Results and notions whose dimension is a parameter of their own statement, namely Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost, The Quadratic Wasserstein Distance on Euclidean Space, Optimal Coupling of Two Probability Measures with Finite Second Moment, Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures and Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound, are applied with $dN$ in place of their $d$ where the measures live on $\mathbb{R}^{dN}$, and The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class, which adopts Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, is applied at the configuration level there; the results on particle blocks, tensor powers and one-particle marginals are read with their dimension parameter equal to $d$, as fixed in N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles. We write $\mathrm{id}$ for the identity map of $\mathbb{R}^{d}$ and $\mathrm{id}_{N}$ for the identity map of $\mathbb{R}^{dN}$, which is the map written $\mathrm{id}$ in clause 3 of the statement; both are Borel, being continuous.

Step 1 (The displacement of the product map). The optimal map $T$ is Borel with $T_{\#}\sigma^{[1]}=\hat{m}$ by Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §map, so its product map $T^{\oplus}:\mathbb{R}^{dN}\to\mathbb{R}^{dN}$ is Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map. Let $g=\mathrm{id}-T:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $g(y)=y-T(y)$. Each component of $g$ is a difference of Borel real functions, hence Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, so $g$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps; by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable (with $\sigma^{[1]}$, $\hat{m}$ and $T$ in place of $\mu$, $\nu$ and $S$) its class is the element $\mathrm{id}-T$ of $L^{2}(\sigma^{[1]};\mathbb{R}^{d})$ named in the statement.

For $x\in\mathbb{R}^{dN}$ and $k\in[N]$, the linearity of $\mathfrak{p}_{k}$ in Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear and the identity $\mathfrak{p}_{k}\circ T^{\oplus}=T\circ\mathfrak{p}_{k}$ of Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map give

Since every point of $\mathbb{R}^{dN}$ is the configuration of its particles (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear), the definition of the product map in Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map yields

Hence, by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §product-maps applied to $g$ and $P=\sigma$, and by the transport cost of an optimal map in The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §cost (with $\sigma^{[1]}$ and $\hat{m}$ in place of $\mu$ and $\nu$, both in $\mathcal{P}_{2}(\mathbb{R}^{d})$),

Step 2 (Clause 1). The push-forward $\sigma_{T}=(T^{\oplus})_{\#}\sigma$ is a probability measure on $\mathbb{R}^{dN}$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. By Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §pushforward (with $h=T$, $p=d$ and $P=\sigma$),

By The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable, applied at the configuration level with $\sigma$ and $T^{\oplus}$ in place of $\mu$ and $S$, then Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §product-maps applied to $T$, and finally The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable at the particle dimension with $\sigma^{[1]}$, $\hat{m}$ and $T$ in place of $\mu$, $\nu$ and $S$,

so $\sigma_{T}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$. Now let $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$. By Tensor Powers Scale the Wasserstein Distance by the Square Root of N, and the One-Particle Marginal is Lipschitz with Constant N^{-1/2} §marginal with $P'=\sigma_{T}$, and $\sigma_{T}^{[1]}=\hat{m}$,

This is clause 1.

Step 3 (Clause 2). Let $\gamma=(\mathrm{id}_{N},T^{\oplus})_{\#}\sigma$. Since $(\mathrm{id}_{N})_{\#}\sigma=\sigma$, the preimage of a set under $\mathrm{id}_{N}$ being the set itself, Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §pushforward (with $\mathrm{id}_{N}$ and $T^{\oplus}$ in place of $S$ and $T$) gives $\gamma\in\Pi(\sigma,\sigma_{T})$ and, with (2),

Both $\sigma$ and $\sigma_{T}$ lie in $\mathcal{P}_{2}(\mathbb{R}^{dN})$ (Step 2), so The Quadratic Wasserstein Distance on Euclidean Space §distance gives $W_{2}(\sigma,\sigma_{T})^{2}\le I(\gamma)$, while clause 1 with $P=\sigma$ gives $N\,W_{2}(\sigma^{[1]},\hat{m})^{2}\le W_{2}(\sigma,\sigma_{T})^{2}$. Thus

and antisymmetry of the order of $\mathbb{R}$ gives $N\,W_{2}(\sigma^{[1]},\hat{m})^{2}=W_{2}(\sigma,\sigma_{T})^{2}=I(\gamma)$. The last equality says that $\gamma$ is optimal in the sense of Optimal Coupling of Two Probability Measures with Finite Second Moment §optimal. This is clause 2.

Step 4 (Clause 3). Suppose that the ordered pair $(\sigma,\sigma_{T})$ is uniquely mapped and that $S$ is an optimal map from $\sigma$ to $\sigma_{T}$. The map $T^{\oplus}$ is Borel (Step 1), satisfies $(T^{\oplus})_{\#}\sigma=\sigma_{T}$ by definition of $\sigma_{T}$, and the coupling $(\mathrm{id}_{N},T^{\oplus})_{\#}\sigma=\gamma$ is optimal by Step 3; so $T^{\oplus}$ is an optimal map from $\sigma$ to $\sigma_{T}$ in the sense of Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §map, read at the configuration level. By The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §unique, applied at the configuration level with $\sigma$, $\sigma_{T}$, $S$ and $T^{\oplus}$ in place of $\mu$, $\nu$, $T$ and $T'$, the classes of $\mathrm{id}_{N}-S$ and of $\mathrm{id}_{N}-T^{\oplus}$ in $L^{2}(\sigma;\mathbb{R}^{dN})$ are equal. By (1) the map $\mathrm{id}_{N}-T^{\oplus}$ is the product map $g^{\oplus}$ of the Borel representative $g$ of the class $\mathrm{id}-T\in L^{2}(\sigma^{[1]};\mathbb{R}^{d})$ (Step 1), so by Product Fields and the Projection onto One-Particle Tangent Fields §product-field its class is the product field $(\mathrm{id}-T)^{\oplus}\in L^{2}(\sigma;\mathbb{R}^{dN})$. Therefore $\mathrm{id}-S=(\mathrm{id}-T)^{\oplus}$ in $L^{2}(\sigma;\mathbb{R}^{dN})$, which is clause 3.