Strong and Weak Convergence of Vector Fields Along Couplings of Vanishing Cost

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $(\nu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}_{2}(\mathbb{R}^{d})$. Limits of sequences of real numbers are those of that definition. For $n\in\mathbb{N}$, $\pi\in\Pi(\nu_{n},\nu)$, $q'\in L^{2}(\nu_{n};\mathbb{R}^{d})$ and $\eta\in L^{2}(\nu;\mathbb{R}^{d})$, the cost $I(\pi)$ and the discrepancy $\int_{\mathbb{R}^{d+d}}\lVert q'(x)-\eta(y)\rVert^{2}\,\pi(dz)$ are the nonnegative real numbers of The Intrinsic Calculus on the Wasserstein Space: Standing Notation (i), and $\mathcal{K}(q',\eta,\pi)$ is the cross pairing of $q'$ and $\eta$ along $\pi$.

1. (Couplings of vanishing cost)(i) A sequence $(\pi_{n})_{n\in\mathbb{N}}$ with $\pi_{n}\in\Pi(\nu_{n},\nu)$ for every $n\in\mathbb{N}$ is a sequence of couplings of vanishing cost from $(\nu_{n})_{n\in\mathbb{N}}$ to $\nu$ if $\lim_{n\to\infty}I(\pi_{n})=0$.

2. (Strong convergence along couplings)(ii) Let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence of couplings of vanishing cost from $(\nu_{n})_{n\in\mathbb{N}}$ to $\nu$, let $q_{n}\in L^{2}(\nu_{n};\mathbb{R}^{d})$ for every $n\in\mathbb{N}$, and let $q\in L^{2}(\nu;\mathbb{R}^{d})$. The sequence $(q_{n})_{n\in\mathbb{N}}$ converges strongly to $q$ along $(\pi_{n})_{n\in\mathbb{N}}$ if

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

3. (Weak convergence along couplings)(iii) With $(\pi_{n})_{n\in\mathbb{N}}$, $(q_{n})_{n\in\mathbb{N}}$ and $q$ as in clause 2, the sequence $(q_{n})_{n\in\mathbb{N}}$ converges weakly to $q$ along $(\pi_{n})_{n\in\mathbb{N}}$ if

$$\lim_{n\to\infty}\mathcal{K}(q_{n},\eta,\pi_{n})=\langle q,\eta\rangle_{\nu}\qquad\text{for every }\eta\in L^{2}(\nu;\mathbb{R}^{d}).$$