Displacement Convexity of a Penalty Pair on the Wasserstein Space

In the setting of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\mu\in\mathcal{D}_{\Sigma}$ and $\nu\in\mathcal{D}$, and let $\pi$ be an optimal coupling of $\mu$ and $\nu$, the couplings $\Pi(\mu,\nu)$ with their quadratic cost $I$ being those of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation §dimensions. Plans, the coordinate projections $\mathrm{pr}_{1},\mathrm{pr}_{2}$ of $\mathbb{R}^{d+d}$, the variables $x,p$ and the integral notation are as fixed there.

(The displacement plan)¶ The displacement plan of $\pi$ is the push-forward

formed with the pairing of $\mathrm{pr}_{1}$ with the pointwise difference $\mathrm{pr}_{2}-\mathrm{pr}_{1}$, a Borel map. It is a plan with first marginal $\mu$. Indeed, by claim 3 of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space the value at $z$ of the nonnegative Borel function $w\mapsto\lVert w\rVert^{2}$ composed with $(\mathrm{pr}_{1},\mathrm{pr}_{2}-\mathrm{pr}_{1})$ is $\lVert\mathrm{pr}_{1}(z)\rVert^{2}+\lVert\mathrm{pr}_{2}(z)-\mathrm{pr}_{1}(z)\rVert^{2}$; so the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward and the additivity of the integral of nonnegative functions in claim 1 of Linearity and Monotonicity of the Lebesgue Integral give

the first integral by the change-of-variables formula and Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling, the second by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost together with the identity $\lVert\mathrm{pr}_{2}(z)-\mathrm{pr}_{1}(z)\rVert=\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert$, which holds because both sides are the Euclidean distance between the two points by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and that distance is symmetric by Euclidean Distance is a Metric on $\mathbb{R}^n$; and $I(\pi)$ is finite by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite, so $\pi^{-}\in\mathcal{P}_{2}(\mathbb{R}^{d+d})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space. Finally $\mathrm{pr}_{1}\circ(\mathrm{pr}_{1},\mathrm{pr}_{2}-\mathrm{pr}_{1})=\mathrm{pr}_{1}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, so the first marginal of $\pi^{-}$ is $(\mathrm{pr}_{1})_{\#}\pi=\mu$. Consequently, $\Sigma(\mu)$ being an element of $L^{2}(\mu;\mathbb{R}^{d})$, the integral

is a real number by Plans and Their Velocity Fields: the Marginal, the Composition Isometry, the Pairing, the Velocity Shift and the Lift §pairing.

(Displacement convexity)¶ The penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is displacement convex if

for all $\mu\in\mathcal{D}_{\Sigma}$, all $\nu\in\mathcal{D}$ and every optimal coupling $\pi$ of $\mu$ and $\nu$, with $\pi^{-}$ the displacement plan of $\pi$.