Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level with particle dimension $d=1$ and the identifications of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative. Let $\hat{\mu}\in\mathcal{P}_{2}(\mathbb{R})$ be atomless, with $L^{2}(\hat{\mu};\mathbb{R})$, its inner product $\langle\cdot,\cdot\rangle_{\hat{\mu}}$ and norm $\lVert\cdot\rVert_{\hat{\mu}}$ as in the setting, and let $N\ge1$ be a natural number, regarded as a real number through the canonical map. Let $F(s)=\hat{\mu}((-\infty,s])$ for $s\in\mathbb{R}$, and for $i\in[N]$ let

the quantile blocks of $\hat{\mu}$ at level $N$. For $x\in\mathbb{R}^{N}$ let $T_{x}:\mathbb{R}\to\mathbb{R}$ be the block map with $T_{x}(s)=x_{i}$ for $s\in B_{i}$. A tuple $x\in\mathbb{R}^{N}$ is ordered if $x_{1}\ge x_{2}\ge\dots\ge x_{N}$; every point of the Weyl chamber $W_{N}$ is ordered. For a Borel $h:\mathbb{R}\to\mathbb{R}$ with $\int|h|\,d\hat{\mu}<\infty$ write $\bar{h}_{i}=N\int_{B_{i}}h\,d\hat{\mu}$ for its block averages, and $m_{i}=\overline{\mathrm{id}}_{i}$.

1. (Blocks)¶ The sets $B_{1},\dots,B_{N}$ are Borel, pairwise disjoint, with union $\mathbb{R}$, and $\hat{\mu}(B_{i})=\frac{1}{N}$ for every $i$; if $i<j$, $s\in B_{i}$ and $t\in B_{j}$, then $t<s$.

2. (Block maps)¶ For $x\in\mathbb{R}^{N}$ the map $T_{x}$ is Borel and nondecreasing when $x$ is ordered, its class lies in $L^{2}(\hat{\mu};\mathbb{R})$, and $(T_{x})_{\#}\hat{\mu}=\mu^{N}_{x}$ is the empirical measure of $x$. For $x,y\in\mathbb{R}^{N}$, $\lVert T_{x}-T_{y}\rVert_{\hat{\mu}}^{2}=\frac{1}{N}\lVert x-y\rVert^{2}$.

3. (Optimality)¶ Let $x\in\mathbb{R}^{N}$ be ordered, let $\nu\in\mathcal{P}_{2}(\mathbb{R})$, and let $T:\mathbb{R}\to\mathbb{R}$ be Borel, nondecreasing on a Borel set of full $\hat{\mu}$-measure, with $T_{\#}\hat{\mu}=\nu$. Then the class of $T$ lies in $L^{2}(\hat{\mu};\mathbb{R})$ and $W_{2}(\mu^{N}_{x},\nu)=\lVert T_{x}-T\rVert_{\hat{\mu}}$. In particular, for ordered $x,y\in\mathbb{R}^{N}$,

4. (The block test function)¶ Let $q\in L^{2}(\hat{\mu};\mathbb{R})$ and $K\in\mathbb{R}$, and let $\chi:\mathbb{R}^{N}\to\mathbb{R}$ be

which does not depend on the representative of $q$. Then $\chi$ is a polynomial of degree at most $2$ in $x$, of class $C^{2}$ on $\mathbb{R}^{N}$, with

$I_{N}$ the identity matrix. For ordered $x$, $\chi(x)=\langle q,T_{x}-\mathrm{id}\rangle_{\hat{\mu}}+K\,W_{2}(\mu^{N}_{x},\hat{\mu})^{2}\ge-\lVert q\rVert_{\hat{\mu}}W_{2}(\mu^{N}_{x},\hat{\mu})+K\,W_{2}(\mu^{N}_{x},\hat{\mu})^{2}$.

5. (Block averages at the particles)¶ Let $q\in L^{2}(\hat{\mu};\mathbb{R})$ and let $h:\mathbb{R}\to\mathbb{R}$ be bounded and Lipschitz with constant $L_{h}\ge0$. Then for every ordered $x\in\mathbb{R}^{N}$