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

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $\mu,\nu_{0}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ be such that the ordered pair $(\mu,\nu_{0})$ is uniquely mapped, and let $T$ be an optimal map from $\mu$ to $\nu_{0}$. Let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}_{2}(\mathbb{R}^{d})$, for each $n\in\mathbb{N}$ let $T_{n}$ be an optimal map from $\mu_{n}$ to $\nu_{0}$, and let $\pi_{n}\in\Pi(\mu_{n},\mu)$ be such that the sequence $(I(\pi_{n}))_{n\in\mathbb{N}}$ converges to $0$. The classes $\mathrm{id}-T\in L^{2}(\mu;\mathbb{R}^{d})$ and $\mathrm{id}-T_{n}\in L^{2}(\mu_{n};\mathbb{R}^{d})$ are the differences of the classes of $\mathrm{id}$ and of the optimal maps fixed in The Intrinsic Calculus on the Wasserstein Space: Standing Notation (ii). Then the following hold.

1. (The optimal maps)(i) The function $z\mapsto\lVert T_{n}(x)-T(y)\rVert^{2}$ on $\mathbb{R}^{d+d}$ is Borel and $\pi_{n}$-integrable for every $n$, and

$$\lim_{n\to\infty}\int_{\mathbb{R}^{d+d}}\bigl\lVert T_{n}(x)-T(y)\bigr\rVert^{2}\,\pi_{n}(dz)=0 .$$

2. (The optimal displacements)(ii) The discrepancy of $\mathrm{id}-T_{n}$ and $\mathrm{id}-T$ along $\pi_{n}$, that clause being read with $\mu_{n}$ in place of its $\nu$, converges to $0$ as $n\to\infty$:

$$\lim_{n\to\infty}\int_{\mathbb{R}^{d+d}}\bigl\lVert\bigl(x-T_{n}(x)\bigr)-\bigl(y-T(y)\bigr)\bigr\rVert^{2}\,\pi_{n}(dz)=0 .$$