Lifted Optimal Maps between Rescaled Heads Converge Strongly to the Noise-Optimal Displacement

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mathcal{P}^{a}_{\rho}$ be the set of The Measures Noise-Connected to the Reference Measure §space and $W_{a}$ the noise Wasserstein distance. For $n\in\mathbb{N}$, the rescaled head map $r_{n}$ and the lift $\Lambda_{n}$ are those of Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space, and for $\lambda\in\mathcal{P}(X)$ we write $\tilde{\lambda}_{n}=(r_{n})_{\#}\lambda$. On $\mathbb{R}^{n}$, the set $\mathcal{P}_{2}(\mathbb{R}^{n})$ of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, the quadratic Wasserstein distance $W_{2}$, optimal couplings and the spaces $L^{2}(\lambda;\mathbb{R}^{n})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu are read with $n$ in place of the dimension written $d$ there, in the settings named there. Let $\mu,\nu\in\mathcal{P}^{a}_{\rho}$ be such that the ordered pair $(\mu,\nu)$ is uniquely noise-mapped, and let $T$ be a noise-optimal map from $\mu$ to $\nu$, with displacement $T-\mathrm{id}\in L^{2}(\mu;X^{a})$, the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields with norm $\lVert\cdot\rVert_{\mu}$. Then $\mu,\nu\in\mathcal{P}_{2}(X)$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §moments, so $\tilde{\mu}_{n},\tilde{\nu}_{n}\in\mathcal{P}_{2}(\mathbb{R}^{n})$ by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §head. For every $n\in\mathbb{N}$ let $S_{n}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a Borel map such that $(\mathrm{id},S_{n})_{\#}\tilde{\mu}_{n}$ is an optimal coupling of $\tilde{\mu}_{n}$ and $\tilde{\nu}_{n}$; write $S_{n}-\mathrm{id}$ for the map $u\mapsto S_{n}(u)-u$.

1. (Head cost)¶ For every $n\in\mathbb{N}$, $\int_{\mathbb{R}^{n}}\lVert S_{n}(u)-u\rVert^{2}\,\tilde{\mu}_{n}(du)=W_{2}(\tilde{\mu}_{n},\tilde{\nu}_{n})^{2}\le W_{a}(\mu,\nu)^{2}$.

2. (The lift)¶ For every $n\in\mathbb{N}$, the lift $\Lambda_{n}(S_{n}-\mathrm{id})\in L^{2}(\mu;X^{a})$ of Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §lift is defined, and $\lVert\Lambda_{n}(S_{n}-\mathrm{id})\rVert_{\mu}\le W_{a}(\mu,\nu)$.

3. (Strong convergence)¶ $\lVert\Lambda_{n}(S_{n}-\mathrm{id})-(T-\mathrm{id})\rVert_{\mu}\to0$ as $n\to\infty$.