Mean-Square Stability of the Optimal Map on the Space of Square-Integrable Random Vectors

In the setting of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation, let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, 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 $\nu$.

1. (Composition with the optimal map)(i) Let $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$. For every representative of $X$ the composition $T\circ X$ is a random vector, its class in $L^{2}(\Omega;\mathbb{R}^{d})$ does not depend on the representative and is again written $T\circ X$, and

$$\lVert T\circ X\rVert_{L^{2}}^{2}=M_{2}(\nu),\qquad\mathcal{L}(T\circ X)=\nu,$$

where $M_{2}(\nu)$ is the second moment of $\nu$.

2. (Mean-square stability)(ii) Let $(X_{n})_{n\in\mathbb{N}}$ and $(Y_{n})_{n\in\mathbb{N}}$ be sequences in $L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X_{n})=\mu$ and $\mathcal{L}(Y_{n})=\nu$ for every $n\in\mathbb{N}$, and suppose that the sequence $(\lVert X_{n}-Y_{n}\rVert_{L^{2}})_{n\in\mathbb{N}}$ converges to $W_{2}(\mu,\nu)$. Then $(\lVert T\circ X_{n}-Y_{n}\rVert_{L^{2}})_{n\in\mathbb{N}}$ converges to $0$.