Intrinsic Test Functions at a Maximiser of the Wasserstein-Doubled Difference

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a Wasserstein-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ whose penalty domain $\mathcal{D}$ has the map property. Upper and lower semicontinuity, and local maxima and local minima relative to $\mathcal{D}$, are understood in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions. Intrinsic test functions $\varphi$ on $\mathcal{D}$, their gradients along couplings $\nabla\varphi(\rho)\in L^{2}(\rho;\mathbb{R}^{d})$ and their translation Hessians $H_{\varphi}(\rho)\in\mathcal{S}(d)$ at $\rho\in\mathcal{D}$ are those of that definition; the cost $I(\pi)$ of a coupling and the discrepancy of two fields along it are those of The Intrinsic Calculus on the Wasserstein Space: Standing Notation §couplings; and that a pair of members of $\mathcal{S}(d)$ is admitted at $\alpha$ is the condition of that clause.

Let $u,v:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ be such that $u$ is upper semicontinuous and $v$ lower semicontinuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ relative to $\mathcal{P}_{2}(\mathbb{R}^{d})$, and $u(\sigma)\le b$ and $b'\le v(\sigma)$ for every $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Let $\delta\in\mathbb{R}$ be positive; by Basic Properties of a Wasserstein-Coercive Penalty Pair §envelopes the $\delta$-envelopes $u^{-}_{\delta}$ of $u$ and $v^{+}_{\delta}$ of $v$ are defined and satisfy $u^{-}_{\delta}=u-\delta\mathcal{E}$ and $v^{+}_{\delta}=v+\delta\mathcal{E}$ on $\mathcal{D}$. Let $\alpha\in\mathbb{R}$ be positive, let $\Psi:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ have 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 $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ satisfy $\Psi(\mu,\nu)\le\Psi(\hat{\mu},\hat{\nu})$ for all $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$.

Then there are $\rho^{*},\sigma^{*}\in\mathcal{D}$ and $\mathbb{X},\mathbb{Y}\in\mathcal{S}(d)$ with the properties below. Since $\mathcal{D}$ has the map property and $\rho^{*},\sigma^{*}\in\mathcal{D}$, both ordered pairs $(\rho^{*},\sigma^{*})$ and $(\sigma^{*},\rho^{*})$ are uniquely mapped; $S$ denotes an optimal map from $\rho^{*}$ to $\sigma^{*}$ and $S'$ an optimal map from $\sigma^{*}$ to $\rho^{*}$, and the classes $\mathrm{id}-S\in L^{2}(\rho^{*};\mathbb{R}^{d})$ and $\mathrm{id}-S'\in L^{2}(\sigma^{*};\mathbb{R}^{d})$ 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)¶ $\Psi(\rho^{*},\sigma^{*})=\Psi(\hat{\mu},\hat{\nu})$.

2. (Admitted matrices)¶ The pair $(\mathbb{X},\mathbb{Y})$ is admitted at $\alpha$.

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