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. 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 (i). 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 (i); and that a pair of members of $\mathcal{S}(d)$ is admitted at $\alpha$ is the condition of that clause.

Let $u,v:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $u(\sigma)\le b$ and $b'\le v(\sigma)$ for every $\sigma\in\mathcal{D}$. Let $\delta\in\mathbb{R}$ be positive; by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Closed Penalty Pair (i), $u$ has penalty-subordinate growth from above and $v$ from below, so that the $\delta$-envelopes $u^{-}_{\delta}$ of $u$ and $v^{+}_{\delta}$ of $v$, functions on $\mathcal{D}$, are defined. Let $\alpha\in\mathbb{R}$ be positive, let $\Psi:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ have value

$$\Psi(\mu,\nu)=u^{-}_{\delta}(\mu)-v^{+}_{\delta}(\nu)-\tfrac{\alpha}{2}\,W_{2}(\mu,\nu)^{2},$$

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 (i) 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 (iii).

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

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

3. (Test functions for the subsolution side)(iii) 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

$$I(\pi)<\varepsilon^{2},\qquad\bigl|u^{-}_{\delta}(\rho)-u^{-}_{\delta}(\rho^{*})\bigr|<\varepsilon,\qquad\int_{\mathbb{R}^{d+d}}\bigl\lVert\nabla\varphi(\rho)(x)-\alpha\bigl(y-S(y)\bigr)\bigr\rVert^{2}\,\pi(dz)<\varepsilon^{2},\qquad\lVert H_{\varphi}(\rho)-\mathbb{X}\rVert<\varepsilon .$$

4. (Test functions for the supersolution side)(iv) 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

$$I(\gamma)<\varepsilon^{2},\qquad\bigl|v^{+}_{\delta}(\sigma)-v^{+}_{\delta}(\sigma^{*})\bigr|<\varepsilon,\qquad\int_{\mathbb{R}^{d+d}}\bigl\lVert\nabla\psi(\sigma)(x)-\alpha\bigl(S'(y)-y\bigr)\bigr\rVert^{2}\,\gamma(dz)<\varepsilon^{2},\qquad\lVert H_{\psi}(\sigma)-\mathbb{Y}\rVert<\varepsilon .$$