Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through Tensor Powers

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a Wasserstein-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ a Wasserstein-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{dN})$, read at the configuration level, whose penalty domains $\mathcal{D}$ and $\mathcal{D}_{N}$ have the map property, and assume that $\mu^{\otimes N}\in\mathcal{D}_{N}$ for every $\mu\in\mathcal{D}$. Intrinsic test functions, their gradients along couplings and translation Hessians, local maxima and minima relative to a penalty domain, the cost $I(\pi)$ of a coupling and the discrepancy of two fields along it are those of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space, read at the particle dimension on $\mathcal{D}$ and at the configuration level on $\mathcal{D}_{N}$; that a pair of symmetric matrices is admitted at $\alpha$ is the condition of that clause, in $\mathcal{S}(d)$ or, at the configuration level, in $\mathcal{S}(dN)$; $a^{\oplus}\in\mathbb{R}^{dN}$ is the diagonal point of $a\in\mathbb{R}^{d}$; and for $Q\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $\Pi_{Q}$ is the one-particle projection, with values in $T_{Q^{[1]}}$.

Let $U:\mathcal{D}_{N}\to\mathbb{R}$, $v:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $U(P)\le b$ for every $P\in\mathcal{D}_{N}$ and $b'\le v(\mu)$ for every $\mu\in\mathcal{D}$. Let $\delta\in\mathbb{R}$ be positive; by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth the $\delta$-envelope $U^{-}_{\delta}$ of $U$ relative to the configuration-level pair and the $\delta$-envelope $v^{+}_{\delta}$ of $v$ relative to the particle-level pair are defined. Let $\alpha\in\mathbb{R}$ be positive, let $\Psi:\mathcal{D}_{N}\times\mathcal{D}\to\mathbb{R}$ have the value

where $\tfrac{\alpha}{2}$ is the product of $\alpha$ with the multiplicative inverse of $2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field), and let $(\hat{P},\hat{\mu})\in\mathcal{D}_{N}\times\mathcal{D}$ satisfy $\Psi(P,\mu)\le\Psi(\hat{P},\hat{\mu})$ for all $(P,\mu)\in\mathcal{D}_{N}\times\mathcal{D}$.

Then there are $\rho^{*}\in\mathcal{D}_{N}$, $\sigma^{*}\in\mathcal{D}$, $\mathbb{X}\in\mathcal{S}(dN)$ and $\mathbb{Y}\in\mathcal{S}(d)$ with the properties below. Put $Q^{*}=(\sigma^{*})^{\otimes N}\in\mathcal{D}_{N}$, so that $(Q^{*})^{[1]}=\sigma^{*}$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor. Since $\mathcal{D}_{N}$ has the map property, both ordered pairs $(\rho^{*},Q^{*})$ and $(Q^{*},\rho^{*})$ are uniquely mapped; $S$ denotes an optimal map from $\rho^{*}$ to $Q^{*}$ and $S'$ an optimal map from $Q^{*}$ to $\rho^{*}$, and the classes $\mathrm{id}-S\in L^{2}(\rho^{*};\mathbb{R}^{dN})$ and $S'-\mathrm{id}\in L^{2}(Q^{*};\mathbb{R}^{dN})$ are supplied by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable and do not depend on these choices by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §unique.

1. (A maximising pair)¶ $\Psi(\rho^{*},\sigma^{*})=\Psi(\hat{P},\hat{\mu})$.

2. (Admitted matrices across the levels)¶ There is $\mathbb{Y}_{N}\in\mathcal{S}(dN)$ such that the pair $(\mathbb{X},\mathbb{Y}_{N})$ is admitted at $\alpha$ at the configuration level and $a^{\oplus}\cdot(\mathbb{Y}_{N}a^{\oplus})=N\,a\cdot(\mathbb{Y}a)$ for every $a\in\mathbb{R}^{d}$.

3. (Test functions for the subsolution side)¶ For every positive $\varepsilon\in\mathbb{R}$ there are $\rho\in\mathcal{D}_{N}$, an intrinsic test function $\varphi$ on $\mathcal{D}_{N}$ at the configuration level such that the function $\mathcal{D}_{N}\to\mathbb{R}$ with value $U^{-}_{\delta}(\rho')-\varphi(\rho')$ at $\rho'$ has a local maximum relative to $\mathcal{D}_{N}$ at $\rho$, and $\pi\in\Pi(\rho,\rho^{*})$ with

4. (Test functions for the supersolution side)¶ For every positive $\varepsilon\in\mathbb{R}$ there are $\sigma\in\mathcal{D}$, an intrinsic test function $\psi$ on $\mathcal{D}$ such that the function $\mathcal{D}\to\mathbb{R}$ with value $v^{+}_{\delta}(\sigma')-\psi(\sigma')$ at $\sigma'$ has a local minimum relative to $\mathcal{D}$ at $\sigma$, and $\gamma\in\Pi(\sigma,\sigma^{*})$ with