N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level

This setting fixes the standing notation used by results on systems of $N$ particles in $\mathbb{R}^{d}$ and on the Wasserstein space of their configuration space. It is layered on The Intrinsic Calculus on the Wasserstein Space: Standing Notation, whose notation is in force throughout; it introduces no new concept and asserts nothing beyond the identifications recorded below, each of which is justified by the reference attached to it.

1. (Particles and configurations)¶ $N\in\mathbb{N}$ is the number of particles, and the dimension $d$ of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data is the particle dimension; $\mathbb{R}^{dN}$ is the configuration space. The block maps $\mathfrak{p}_{k}:\mathbb{R}^{dN}\to\mathbb{R}^{d}$, configurations $[y_{1},\dots,y_{N}]$, product maps $h^{\oplus}$ and diagonal points $a^{\oplus}$ are those of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks, Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration, Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map and Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §diagonal; tensor powers $\rho^{\otimes N}$, one-particle marginals $P^{[1]}$ and empirical measures $\mu^{N}_{x}$ are those of The Tensor Power of a Probability Measure on Euclidean Space §tensor, The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal and The Empirical Measure of a Configuration of N Particles §empirical. All of them are read with their dimension parameter, written $q$ in those items, equal to $d$, and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts, Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts, Tensor Powers Scale the Wasserstein Distance by the Square Root of N, and the One-Particle Marginal is Lipschitz with Constant N^{-1/2} and Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration are in force with that parameter equal to $d$; in a result adopting this setting the letter $q$ keeps the meaning fixed in The Intrinsic Calculus on the Wasserstein Space: Standing Notation §operators. In a result adopting this setting the letters $P$ and $P'$ denote probability measures on the configuration space, and the probability space $(\Omega,\mathcal{F},P)$ of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data is not used.

2. (The configuration level)¶ The dimension fixed in The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data is an arbitrary natural number $d$ with $1\le d$. Hence each result adopting a setting that takes its dimension from that clause, in particular The Intrinsic Calculus on the Wasserstein Space: Standing Notation, its earlier version The Intrinsic Calculus on the Wasserstein Space: Standing Notation, or a setting on which either is layered, holds for every natural number in place of $d$, and each notion fixed by such a setting, or by a definition adopting one, is defined for every natural number in place of $d$. A result adopting the present setting may apply such a result, or use such a notion, at the configuration level, that is, read with the natural number $dN$ in place of $d$; here $1\le dN$ by claim 4 of Properties of the Order on the Natural Numbers. Objects formed at the configuration level carry the dimension explicitly: $\mathcal{P}_{2}(\mathbb{R}^{dN})$ with the distance $W_{2}$, the spaces $L^{2}(P;\mathbb{R}^{dN})$ and tangent spaces $T_{P}$ for $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, the symmetric matrices $\mathcal{S}(dN)$, penalty pairs on $\mathcal{P}_{2}(\mathbb{R}^{dN})$ with translation Hessians in $\mathcal{S}(dN)$, second-order equation operators over subsets of $\mathcal{P}_{2}(\mathbb{R}^{dN})$, intrinsic test functions on such subsets, and classical and viscosity sub- and supersolutions. A result or notion used without this qualification is read at the particle dimension $d$.