Stability of the Optimal Map and of the Displacement on the Torus Along Couplings of Nearly Optimal Cost

In the setting of The Flat Torus: Standing Notation, used with $n=d$ for a natural number $d$ with $1\le d$, and of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $\varpi$ be the wrapped displacement and $d_{\mathbb{T}}$ the flat torus distance; let $\mathcal{P}(\mathbb{T}^{d})$, the torus cost $I_{\mathbb{T}}$ and the torus Wasserstein distance $W_{\mathbb{T}}$ be as in Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures. Let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ be such that $(\mu,\nu)$ is uniquely mapped on the torus, let $T$ be an optimal map on the torus from $\mu$ to $\nu$ with displacement field $v_{T}$, and let $(\gamma_{j})_{j\in\mathbb{N}}$ be a sequence in $\Pi(\mu,\nu)$ such that $(I_{\mathbb{T}}(\gamma_{j}))_{j\in\mathbb{N}}$ converges to $W_{\mathbb{T}}(\mu,\nu)^{2}$.

The functions $w\mapsto d_{\mathbb{T}}(T(\mathrm{pr}_{1}(w)),\mathrm{pr}_{2}(w))^{2}$ and $w\mapsto\lVert\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))-v_{T}(\mathrm{pr}_{1}(w))\rVert^{2}$ on $\mathbb{R}^{d+d}$ are Borel, by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz, Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, and bounded by $d/4$ and $d$ respectively, by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range and the inequality $\lVert a-b\rVert^{2}\le2\lVert a\rVert^{2}+2\lVert b\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions; so their integrals against each $\gamma_{j}$ are real numbers.

1. (Stability of the map)¶ The sequence $\bigl(\int_{\mathbb{R}^{d+d}}d_{\mathbb{T}}(T(\mathrm{pr}_{1}(w)),\mathrm{pr}_{2}(w))^{2}\,\gamma_{j}(dw)\bigr)_{j\in\mathbb{N}}$ converges to $0$.

2. (Stability of the displacement)¶ If moreover $\mu$ is absolutely continuous, then the sequence $\bigl(\int_{\mathbb{R}^{d+d}}\lVert\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))-v_{T}(\mathrm{pr}_{1}(w))\rVert^{2}\,\gamma_{j}(dw)\bigr)_{j\in\mathbb{N}}$ converges to $0$.