Mean-Square Stability of the Optimal Map of a Uniquely Mapped Pair Along Couplings of Nearly Optimal Cost

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$ and let $\mu,\nu$ belong to the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment, the second moment of $\nu$ being written $M_{2}(\nu)$. Let $\Pi(\mu,\nu)$ be the set of their couplings, let $I$ be the quadratic cost, and let $W_{2}(\mu,\nu)$ be the quadratic Wasserstein distance. Suppose that the ordered pair $(\mu,\nu)$ is uniquely mapped, and let $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be an optimal map from $\mu$ to $\nu$.

For $\pi\in\Pi(\mu,\nu)$ the function

$$D_{T}(z)=\bigl\lVert T(\mathrm{pr}_{1}(z))-\mathrm{pr}_{2}(z)\bigr\rVert^{2}\qquad(z\in\mathbb{R}^{d+d})$$

is Borel and nonnegative by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), applied to the Borel maps $T\circ\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$, and $\int_{\mathbb{R}^{d+d}}D_{T}\,d\pi$ is a real number not exceeding $4M_{2}(\nu)$: by the inequality $\lVert a-b\rVert^{2}\le2\lVert a\rVert^{2}+2\lVert b\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) it is dominated by $2\lVert T(\mathrm{pr}_{1}(z))\rVert^{2}+2\lVert\mathrm{pr}_{2}(z)\rVert^{2}$, whose integral against $\pi$ is $2M_{2}(\nu)+2M_{2}(\nu)$ by the change-of-variables formula together with $(\mathrm{pr}_{1})_{\#}\pi=\mu$, $(\mathrm{pr}_{2})_{\#}\pi=\nu$ and $T_{\#}\mu=\nu$.

1. (Mean-square stability)(i) Let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence in $\Pi(\mu,\nu)$ such that $(I(\pi_{n}))_{n\in\mathbb{N}}$ converges to $W_{2}(\mu,\nu)^{2}$. Then the sequence of real numbers $\bigl(\int_{\mathbb{R}^{d+d}}D_{T}\,d\pi_{n}\bigr)_{n\in\mathbb{N}}$ converges to $0$.