Optimal Transport on the Flat Torus: Standing Notation

This setting fixes the standing notation for optimal transport on the flat torus. It introduces no new concept and asserts nothing beyond the identifications recorded below, each justified by the reference attached to it.

1. (Conventions)¶ $d$ denotes a natural number with $1\le d$. The notation of The Flat Torus: Standing Notation, used with $n=d$, and of Probability Measures on Euclidean Space and Random Vectors: Standing Notation is in force: in particular the half-open unit cell $Q$ and the wrapping map $\pi$, the integer lattice $\mathbb{Z}^{d}$, the periodic classes $C_{\mathrm{per}}$ and $C^{k}_{\mathrm{per}}$, Borel maps, $\mathcal{P}(\mathbb{R}^{q})$ and integrals, push-forwards and the projections $\mathrm{pr}_{1},\mathrm{pr}_{2}$ of $\mathbb{R}^{d+d}$. $\mathrm{id}$ is the identity map of $\mathbb{R}^{d}$.

2. (Torus distance)¶ $\varpi$ is the wrapped displacement and $d_{\mathbb{T}}$ the flat torus distance; a point $z\in\mathbb{R}^{d}$ is regular in the sense of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §regular-point. Couplings are written $\gamma$; the letter $\pi$ is reserved for the wrapping map.

3. (Measures, couplings and distance)¶ $\mathcal{P}(\mathbb{T}^{d})$ is the set of probability measures on the torus; for $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$, $\Pi(\mu,\nu)$ is the set of their couplings, $I_{\mathbb{T}}$ the torus cost, $W_{\mathbb{T}}$ the torus Wasserstein distance, and optimal couplings are those of Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §optimal. By The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §metric, $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$ is a metric space, and convergence of sequences in $\mathcal{P}(\mathbb{T}^{d})$ refers to it unless stated otherwise.

4. (Maps)¶ Absolutely continuous measures are those of Absolutely Continuous Probability Measure on Euclidean Space §ac; optimal maps, uniquely mapped pairs and the displacement field $v_{T}$ of a Borel map $T$ are those of Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §map, Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §uniquely-mapped and Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement, always understood on the torus.

5. (Vector fields)¶ For $\mu\in\mathcal{P}(\mathbb{R}^{d})$, $L^{2}(\mu;\mathbb{R}^{d})$ is the space of classes of square-integrable random vectors in $\mathbb{R}^{d}$ on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$; it is a real Hilbert space by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations §hilbert. Its inner product and norm are written $\langle\xi,\eta\rangle_{\mu}=\int\xi\cdot\eta\,d\mu$ and $\lVert\xi\rVert_{\mu}$, closures in it are taken for that norm, and a class and a representative are denoted by the same symbol, as in The Space of Square-Integrable Random Vectors §convention.

6. (Periodic calculus)¶ For $f\in C^{1}_{\mathrm{per}}$, $\nabla f$ is the gradient; its components are continuous and periodic by Elementary Properties of Lattice-Periodic Functions §derivative, hence bounded by Elementary Properties of Lattice-Periodic Functions §bounded, so $\nabla f$ is Borel with $\int\lVert\nabla f\rVert^{2}\,d\mu<\infty$ and its class belongs to $L^{2}(\mu;\mathbb{R}^{d})$ for every $\mu\in\mathcal{P}(\mathbb{R}^{d})$. For $f\in C^{2}_{\mathrm{per}}$, $\Delta f$ is its Laplacian, a continuous periodic function by the same two clauses, hence bounded, Borel and integrable with respect to every $\mu\in\mathcal{P}(\mathbb{R}^{d})$.

7. (Background)¶ The following results are in force by reference: The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound, The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions, The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist, The Support of an Optimal Coupling on the Torus is Torus-Cyclically Monotone, A Periodic Potential for a Torus-Cyclically Monotone Set: Lipschitz Periodic Part, Convex Lift and Subgradients, McCann's Theorem on the Flat Torus: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map with a Periodic Potential, Stability of the Optimal Map and of the Displacement on the Torus Along Couplings of Nearly Optimal Cost, Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound, Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling and The Support of a Borel Measure is Closed, and Carries Full Measure on a Separable Space.