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

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. Optimal maps and uniquely mapped ordered pairs are those of the setting, read at the particle dimension and at the configuration level; optimal couplings are those of that definition; $h^{\oplus}$ is the product map of a map $h:\mathbb{R}^{d}\to\mathbb{R}^{d}$, and $g^{\oplus}\in L^{2}(P;\mathbb{R}^{dN})$ is the product field of $g\in L^{2}(P^{[1]};\mathbb{R}^{d})$. For $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $P^{[1]}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments.

Let $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ and $\hat{m}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ be such that the ordered pair $(\sigma^{[1]},\hat{m})$ is uniquely mapped, let $T$ be an optimal map from $\sigma^{[1]}$ to $\hat{m}$, with class $\mathrm{id}-T\in L^{2}(\sigma^{[1]};\mathbb{R}^{d})$ as in The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable, and put $\sigma_{T}=(T^{\oplus})_{\#}\sigma$.

1. (Majorant)¶ $\sigma_{T}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $\sigma_{T}^{[1]}=\hat{m}$, and

2. (Touching)¶ $N\,W_{2}(\sigma^{[1]},\hat{m})^{2}=W_{2}(\sigma,\sigma_{T})^{2}$, and the coupling $(\mathrm{id},T^{\oplus})_{\#}\sigma$ of $\sigma$ and $\sigma_{T}$ is optimal.

3. (The optimal map is the product map)¶ If the ordered pair $(\sigma,\sigma_{T})$ is uniquely mapped and $S$ is an optimal map from $\sigma$ to $\sigma_{T}$, then $\mathrm{id}-S=(\mathrm{id}-T)^{\oplus}$ in $L^{2}(\sigma;\mathbb{R}^{dN})$.