A Pair Maximising a Quadratically Penalised Difference of Law-Invariant Functions Within Its Laws Realises an Optimal Coupling

In the setting of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation, assume that $(\Omega,\mathcal{F},P)$ is rich. Let $L^{2}(\Omega;\mathbb{R}^{d})$ be the space of classes of square-integrable random vectors, and $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ the Wasserstein space, so that $\mathcal{L}(X)\in\mathcal{P}_{2}(\mathbb{R}^{d})$ for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$ by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §law-map. Let $U,V:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$ be law-invariant, let $\alpha\in\mathbb{R}$ be positive, and let $X_{0},Y_{0}\in L^{2}(\Omega;\mathbb{R}^{d})$ satisfy

for all $X,Y\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mathcal{L}(X_{0})$ and $\mathcal{L}(Y)=\mathcal{L}(Y_{0})$. Then the following hold.

1. (The penalty is the Wasserstein distance)¶

2. (The pair realises an optimal coupling)¶ The law of the pairing $(X_{0},Y_{0})$ of any representatives of $X_{0}$ and $Y_{0}$, a random vector in $\mathbb{R}^{d+d}$ by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §pair, is an optimal coupling of $\mathcal{L}(X_{0})$ and $\mathcal{L}(Y_{0})$.