Proof of Stability of the Optimal Displacement Under Perturbation of the Source Along Couplings of Vanishing Cost

Each result cited is universally quantified over the data in its own statement. For $z\in\mathbb{R}^{d+d}$ we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$, as in The Intrinsic Calculus on the Wasserstein Space: Standing Notation (i), and $w=W_{2}(\mu,\nu_{0})$, $w_{n}=W_{2}(\mu_{n},\nu_{0})$ and $a_{n}=\sqrt{I(\pi_{n})}$ for $n\in\mathbb{N}$, all nonnegative real numbers. The maps $T$ and $T_{n}$ are Borel, being optimal maps (Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures (i)), and so are the coordinate projections, by claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets; compositions of Borel maps are Borel.

Step 1 (a coupling of $\mu$ with $\nu_{0}$). Fix $n\in\mathbb{N}$ and let $G_{n}:\mathbb{R}^{d+d}\to\mathbb{R}^{d+d}$ be the map with value $(y,T_{n}(x))$ at $z$; its two components $\mathrm{pr}_{2}$ and $T_{n}\circ\mathrm{pr}_{1}$ are Borel, so $G_{n}$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii). Put $\gamma_{n}=(G_{n})_{\#}\pi_{n}$. Since $\mathrm{pr}_{1}\circ G_{n}=\mathrm{pr}_{2}$ and $\mathrm{pr}_{2}\circ G_{n}=T_{n}\circ\mathrm{pr}_{1}$, the definition of the push-forward in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) gives $(\mathrm{pr}_{1})_{\#}\gamma_{n}=(\mathrm{pr}_{2})_{\#}\pi_{n}=\mu$ and $(\mathrm{pr}_{2})_{\#}\gamma_{n}=(T_{n})_{\#}\bigl((\mathrm{pr}_{1})_{\#}\pi_{n}\bigr)=(T_{n})_{\#}\mu_{n}=\nu_{0}$, so $\gamma_{n}\in\Pi(\mu,\nu_{0})$. By the change-of-variables formula of that clause,

$$I(\gamma_{n})=\int_{\mathbb{R}^{d+d}}\bigl\lVert y-T_{n}(x)\bigr\rVert^{2}\,\pi_{n}(dz),\qquad\int_{\mathbb{R}^{d+d}}D_{T}\,d\gamma_{n}=\int_{\mathbb{R}^{d+d}}\bigl\lVert T(y)-T_{n}(x)\bigr\rVert^{2}\,\pi_{n}(dz),$$

where $D_{T}$ is the function of Mean-Square Stability of the Optimal Map of a Uniquely Mapped Pair Along Couplings of Nearly Optimal Cost, whose integral against $\gamma_{n}$ is a real number by the preamble of that theorem. The integrand on the right of the second identity equals $\lVert T_{n}(x)-T(y)\rVert^{2}$ by claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square with the multiplier $-1$, whose absolute value is $1$ by claim 2 of Properties of the Absolute Value in an Ordered Field; it is Borel 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 $T_{n}\circ\mathrm{pr}_{1}$ and $T\circ\mathrm{pr}_{2}$, and $\pi_{n}$-integrable because its integral is the real number just named. This proves the first sentence of claim 1.

Step 2 (the cost of $\gamma_{n}$ tends to $w^{2}$). By The Quadratic Wasserstein Distance on Euclidean Space (i), $w^{2}\le I(\gamma_{n})$ and $W_{2}(\mu_{n},\mu)^{2}\le I(\pi_{n})$, so $W_{2}(\mu_{n},\mu)\le a_{n}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. In the real Hilbert space $L^{2}(\pi_{n};\mathbb{R}^{d})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii) (read with $d+d$ in place of $q$ and $d$ in place of $r$), let $A$ and $B$ be the classes of $z\mapsto y-x$ and $z\mapsto x-T_{n}(x)$, which are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv). Then $\lVert A\rVert_{\pi_{n}}^{2}=I(\pi_{n})$, and, by the change-of-variables formula with $(\mathrm{pr}_{1})_{\#}\pi_{n}=\mu_{n}$ and by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (ii),

$$\lVert B\rVert_{\pi_{n}}^{2}=\int_{\mathbb{R}^{d}}\bigl\lVert x-T_{n}(x)\bigr\rVert^{2}\,\mu_{n}(dx)=\lVert\mathrm{id}-T_{n}\rVert_{\mu_{n}}^{2}=w_{n}^{2}.$$

The class $A+B$ is that of $z\mapsto y-T_{n}(x)$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (v), so $\sqrt{I(\gamma_{n})}=\lVert A+B\rVert_{\pi_{n}}\le a_{n}+w_{n}$ by the triangle inequality of claim 1 of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity. By the triangle inequality and the symmetry of $W_{2}$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space (iii), The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space (i)), $w_{n}\le W_{2}(\mu_{n},\mu)+w\le a_{n}+w$. Hence $\sqrt{I(\gamma_{n})}\le2a_{n}+w$, and by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and the commutativity, associativity and distributivity axioms of Field (expanding $(2a_{n}+w)(2a_{n}+w)$),

$$w^{2}\ \le\ I(\gamma_{n})\ \le\ w^{2}+4a_{n}w+4a_{n}^{2}\qquad(n\in\mathbb{N}).$$

The sequence $(a_{n})$ converges to $0$: given a positive $\varepsilon\in\mathbb{R}$, the positive $\varepsilon^{2}$ exceeds $I(\pi_{n})$ for all large $n$, and then $a_{n}<\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By claims 1 and 3 of Arithmetic of Limits of Real Sequences the right-hand side converges to $w^{2}$, and so $(I(\gamma_{n}))$ converges to $w^{2}$ by the squeeze principle, claim 2 of Order Properties of Limits of Real Sequences.

Step 3 (claim 1). Each $\gamma_{n}$ belongs to $\Pi(\mu,\nu_{0})$, the costs converge to $W_{2}(\mu,\nu_{0})^{2}$, and the pair $(\mu,\nu_{0})$ is uniquely mapped with optimal map $T$. Hence Mean-Square Stability of the Optimal Map of a Uniquely Mapped Pair Along Couplings of Nearly Optimal Cost (i) gives $\int D_{T}\,d\gamma_{n}\to0$, which by Step 1 is the limit asserted in claim 1.

Step 4 (claim 2). The maps $x\mapsto x-T_{n}(x)$ and $y\mapsto y-T(y)$ represent $\mathrm{id}-T_{n}$ and $\mathrm{id}-T$, by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (v), and the discrepancy does not depend on the representatives by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures (i). For every $z$, the vector arithmetic of $\mathbb{R}^{d}$ gives $(x-T_{n}(x))-(y-T(y))=(x-y)-(T_{n}(x)-T(y))$, so the inequality $\lVert s-t\rVert^{2}\le2\lVert s\rVert^{2}+2\lVert t\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) gives

$$\bigl\lVert\bigl(x-T_{n}(x)\bigr)-\bigl(y-T(y)\bigr)\bigr\rVert^{2}\le2\lVert x-y\rVert^{2}+2\bigl\lVert T_{n}(x)-T(y)\bigr\rVert^{2}.$$

All three functions are nonnegative and Borel, so claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion bounds the discrepancy by $2I(\pi_{n})+2\int\lVert T_{n}(x)-T(y)\rVert^{2}\,\pi_{n}(dz)$, which converges to $0$ by the hypothesis, claim 1 and claims 1 and 3 of Arithmetic of Limits of Real Sequences. The discrepancy being nonnegative, it converges to $0$ by claim 2 of Order Properties of Limits of Real Sequences.