Proof of Half the Squared Torus Wasserstein Distance to a Fixed Measure is an Intrinsic Test Function on the Absolutely Continuous Measures

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. Write $\Phi=\Phi_{\nu}$. Integrals over $\mathbb{R}^{d+d}$ are formed in $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\gamma)$ for the coupling $\gamma$ at hand, as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures; by that clause a nonnegative Borel real function is integrated as a $[0,\infty]$-valued map, and for a bounded one this integral is real and equals its integral as an integrable function by claim 6(c) of Borel Measurability and Bounded Integration on a Metric Space. Composites of Borel maps are Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, and the square root is the nonnegative one of Existence and Uniqueness of the Nonnegative Square Root. For nonnegative reals $a,b$ one has $a\le b$ if and only if $a^{2}\le b^{2}$ (and likewise with strict inequalities), because $b^{2}-a^{2}=(b-a)(b+a)$ with $b+a\ge0$, and $b+a=0$ forces $a=b=0$; so the square root preserves the order of nonnegative reals, strict or not.

Preliminaries on $W_{\mathbb{T}}$. Let $\mu,\mu'\in\mathcal{P}(\mathbb{T}^{d})$. (i) By Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost every $\gamma\in\Pi(\mu,\nu)$ has $0\le I_{\mathbb{T}}(\gamma)\le d/4$, and $\Pi(\mu,\nu)$ is nonempty (preamble of Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings); since by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance $W_{\mathbb{T}}(\mu,\nu)^{2}$ is the greatest lower bound of these costs, $W_{\mathbb{T}}(\mu,\nu)^{2}\le d/4$ and $0\le W_{\mathbb{T}}(\mu,\nu)\le\sqrt{d}/2$. (ii) For the same reason, $W_{\mathbb{T}}(\mu,\mu')\le\sqrt{I_{\mathbb{T}}(\gamma)}$ for every $\gamma\in\Pi(\mu,\mu')$. (iii) $W_{\mathbb{T}}$ is a metric on $\mathcal{P}(\mathbb{T}^{d})$ by The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §metric; its triangle inequality and symmetry give $W_{\mathbb{T}}(\mu,\nu)\le W_{\mathbb{T}}(\mu,\mu')+W_{\mathbb{T}}(\mu',\nu)$ and $W_{\mathbb{T}}(\mu',\nu)\le W_{\mathbb{T}}(\mu,\mu')+W_{\mathbb{T}}(\mu,\nu)$, hence, by claim 6 of Properties of the Absolute Value in an Ordered Field,

$$\bigl|W_{\mathbb{T}}(\mu,\nu)-W_{\mathbb{T}}(\mu',\nu)\bigr|\le W_{\mathbb{T}}(\mu,\mu').$$

Step 1 (Clause 1: a composite displacement). Let $\mu,\mu_{n},T,T_{n},\gamma_{n}$ be as in clause 1, write $W=W_{\mathbb{T}}(\mu,\nu)$ and $s_{n}=\sqrt{I_{\mathbb{T}}(\gamma_{n})}$. By Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §map, $T$ and $T_{n}$ are Borel with $T_{\#}\mu=\nu$ and $(T_{n})_{\#}\mu_{n}=\nu$. By the preamble of Half the Squared Torus Wasserstein Distance to a Fixed Measure: the One-Sided Bound Along Couplings, Differentiability at an Absolutely Continuous Source, and the Tangent Gradient, $v_{T}$ and $v_{T_{n}}$ are Borel with $\lVert v_{T}\rVert,\lVert v_{T_{n}}\rVert\le\sqrt{d}/2$, and their classes lie in $L^{2}(\mu;\mathbb{R}^{d})$ and $L^{2}(\mu_{n};\mathbb{R}^{d})$. Fix $n\in\mathbb{N}$ and define on $\mathbb{R}^{d+d}$

$$X=\mathrm{pr}_{2},\qquad Y_{n}=T_{n}\circ\mathrm{pr}_{1},\qquad Z_{n}(w)=v_{T_{n}}(\mathrm{pr}_{1}(w))-\varpi\bigl(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w)\bigr).$$

These maps are Borel: the projections by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, $\varpi$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz, and differences of Borel maps into $\mathbb{R}^{d}$ because their components are differences of Borel real functions (claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). We check the hypotheses of Nearly Optimal Composite Displacements on the Torus Converge to the Optimal Displacement, used with $q=d+d$, $\Sigma_{n}=\gamma_{n}$, $X_{n}=X$, $Y_{n}$ and $Z_{n}$.

(a) Marginals. As $\gamma_{n}\in\Pi(\mu_{n},\mu)$, Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling gives $X_{\#}\gamma_{n}=\mu$, and for $B\in\mathcal{B}(\mathbb{R}^{d})$, with $T_{n}^{-1}(B)\in\mathcal{B}(\mathbb{R}^{d})$,

$$(Y_{n})_{\#}\gamma_{n}(B)=\gamma_{n}\bigl(\mathrm{pr}_{1}^{-1}(T_{n}^{-1}(B))\bigr)=\mu_{n}(T_{n}^{-1}(B))=(T_{n})_{\#}\mu_{n}(B)=\nu(B),$$

by the defining formula of the push-forward in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward.

(b) Integer difference. Let $w\in\mathbb{R}^{d+d}$, $x'=\mathrm{pr}_{1}(w)$ and $x=\mathrm{pr}_{2}(w)$. Since $v_{T_{n}}(x')=\varpi(T_{n}(x')-x')$ by Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement,

$$Y_{n}(w)-X(w)-Z_{n}(w)=\bigl[(T_{n}(x')-x')-\varpi(T_{n}(x')-x')\bigr]-\bigl[(x-x')-\varpi(x-x')\bigr],$$

and both brackets lie in $\mathbb{Z}^{d}$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range; so the difference lies in $\mathbb{Z}^{d}$.

(c) The second moment of $Z_{n}$. Write $\alpha(w)=v_{T_{n}}(\mathrm{pr}_{1}(w))$ and $\beta(w)=\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))$. By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claims 1, 3 and 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$,

$$\lVert Z_{n}(w)\rVert^{2}=\lVert\alpha(w)\rVert^{2}-2\,\alpha(w)\cdot\beta(w)+\lVert\beta(w)\rVert^{2}.$$

The three functions on the right are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions and bounded in absolute value by $d/4$, by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, the bound on $v_{T_{n}}$ and the inequality $|a\cdot b|\le\lVert a\rVert\,\lVert b\rVert$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions; so they are integrable (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures) and $\lVert Z_{n}\rVert^{2}$ is bounded, whence $\int\lVert Z_{n}\rVert^{2}\,d\gamma_{n}<\infty$. By the change-of-variables formula with $(\mathrm{pr}_{1})_{\#}\gamma_{n}=\mu_{n}$ and Optimal Transport on the Flat Torus: Standing Notation §fields, $\int\lVert\alpha\rVert^{2}\,d\gamma_{n}=\int\lVert v_{T_{n}}\rVert^{2}\,d\mu_{n}=\lVert v_{T_{n}}\rVert_{\mu_{n}}^{2}$. By The Torus Displacement Pairing of a Vector Field Along a Coupling §pairing, applied with $\mu_{n}$ and $\mu$ in the roles of $\mu$ and $\nu$ and with the Borel representative $v_{T_{n}}$, $\int\alpha\cdot\beta\,d\gamma_{n}=\mathcal{J}_{\mathbb{T}}(v_{T_{n}},\gamma_{n})$. By The Wrapped Displacement and the Flat Torus Distance §distance, $\lVert\beta(w)\rVert=d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))$, so $\int\lVert\beta\rVert^{2}\,d\gamma_{n}=I_{\mathbb{T}}(\gamma_{n})$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost. Claim 2 of Linearity and Monotonicity of the Lebesgue Integral and The Torus Displacement Pairing: Linearity, the Cost Bound, and Vanishing of a Field with First-Order Small Pairings §bound then give

$$\int_{\mathbb{R}^{d+d}}\lVert Z_{n}\rVert^{2}\,d\gamma_{n}=\lVert v_{T_{n}}\rVert_{\mu_{n}}^{2}-2\,\mathcal{J}_{\mathbb{T}}(v_{T_{n}},\gamma_{n})+s_{n}^{2}\le\lVert v_{T_{n}}\rVert_{\mu_{n}}^{2}+2\lVert v_{T_{n}}\rVert_{\mu_{n}}s_{n}+s_{n}^{2}=\bigl(\lVert v_{T_{n}}\rVert_{\mu_{n}}+s_{n}\bigr)^{2}.$$

By Half the Squared Torus Wasserstein Distance to a Fixed Measure: the One-Sided Bound Along Couplings, Differentiability at an Absolutely Continuous Source, and the Tangent Gradient §upper, applied to $\mu_{n}$ and $T_{n}$, $\tfrac12W_{\mathbb{T}}(\mu_{n},\nu)^{2}=\Phi(\mu_{n})=\tfrac12\lVert v_{T_{n}}\rVert_{\mu_{n}}^{2}$; both $W_{\mathbb{T}}(\mu_{n},\nu)$ and $\lVert v_{T_{n}}\rVert_{\mu_{n}}$ are nonnegative, so they are equal by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. By Preliminary (iii), applied to $\mu_{n}$ and $\mu$, and Preliminary (ii), applied to $\gamma_{n}\in\Pi(\mu_{n},\mu)$, $W_{\mathbb{T}}(\mu_{n},\nu)\le W+W_{\mathbb{T}}(\mu_{n},\mu)\le W+s_{n}$. Hence, as all quantities are nonnegative,

$$\int_{\mathbb{R}^{d+d}}\lVert Z_{n}\rVert^{2}\,d\gamma_{n}\le(W+2s_{n})^{2}=W^{2}+4Ws_{n}+4s_{n}^{2}. \tag{1}$$

(d) Near optimality. Let $\varepsilon$ be a positive real number and let $\delta$ be the smaller of $1$ and $\varepsilon/(2\sqrt{d}+4)$, a positive real. As $(I_{\mathbb{T}}(\gamma_{n}))_{n}$ converges to $0$, Limit of a Sequence of Real Numbers, used with $\delta^{2}$, gives $N\in\mathbb{N}$ with $I_{\mathbb{T}}(\gamma_{n})<\delta^{2}$, hence $0\le s_{n}<\delta\le1$, for every $n\ge N$. For such $n$, $s_{n}^{2}\le s_{n}$ and $W\le\sqrt{d}/2$ (Preliminary (i)), so $(1)$ gives

$$\int_{\mathbb{R}^{d+d}}\lVert Z_{n}\rVert^{2}\,d\gamma_{n}\le W^{2}+(2\sqrt{d}+4)\,s_{n}\le W^{2}+(2\sqrt{d}+4)\,\delta\le W^{2}+\varepsilon .$$

The order of choice is $\varepsilon$, then $\delta$, then $N$.

By (a)-(d), Nearly Optimal Composite Displacements on the Torus Converge to the Optimal Displacement §convergence applies ($\mu$ being absolutely continuous and $T$ an optimal map from $\mu$ to $\nu$): the sequence $c_{n}=\int\lVert Z_{n}-v_{T}\circ\mathrm{pr}_{2}\rVert^{2}\,d\gamma_{n}$ converges to $0$.

Step 2 (Clause 1: conclusion). For $w\in\mathbb{R}^{d+d}$ we have $v_{T_{n}}(\mathrm{pr}_{1}(w))-v_{T}(\mathrm{pr}_{2}(w))=\bigl(Z_{n}(w)-v_{T}(\mathrm{pr}_{2}(w))\bigr)+\beta(w)$, with $\beta$ as in Step 1(c). The first inequality of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, applied to $Z_{n}(w)-v_{T}(\mathrm{pr}_{2}(w))$ and $-\beta(w)$, whose norm is $\lVert\beta(w)\rVert=d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))$ by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, gives

$$\bigl\lVert v_{T_{n}}(\mathrm{pr}_{1}(w))-v_{T}(\mathrm{pr}_{2}(w))\bigr\rVert^{2}\le2\bigl\lVert Z_{n}(w)-v_{T}(\mathrm{pr}_{2}(w))\bigr\rVert^{2}+2\,d_{\mathbb{T}}\bigl(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w)\bigr)^{2}.$$

All functions here are nonnegative and Borel (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions and The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz). Integrating against $\gamma_{n}$ with claim 1 of Linearity and Monotonicity of the Lebesgue Integral,

$$0\le e_{n}:=\int_{\mathbb{R}^{d+d}}\bigl\lVert v_{T_{n}}(\mathrm{pr}_{1}(w))-v_{T}(\mathrm{pr}_{2}(w))\bigr\rVert^{2}\,\gamma_{n}(dw)\le2c_{n}+2I_{\mathbb{T}}(\gamma_{n}).$$

Given a positive real $\varepsilon$, choose $N_{1}$ with $c_{n}<\varepsilon/4$ for $n\ge N_{1}$ (Step 1) and $N_{2}$ with $I_{\mathbb{T}}(\gamma_{n})<\varepsilon/4$ for $n\ge N_{2}$ (hypothesis), and let $N$ be the larger; then $|e_{n}-0|<\varepsilon$ for $n\ge N$. By Limit of a Sequence of Real Numbers, $(e_{n})_{n\in\mathbb{N}}$ converges to $0$, which is clause 1.

Step 3 (Clause 2, property (a): continuity). Let $\mu,\mu'\in\mathcal{P}(\mathbb{T}^{d})$ and write $W_{1}=W_{\mathbb{T}}(\mu,\nu)$, $W_{2}=W_{\mathbb{T}}(\mu',\nu)$. Then $W_{1}^{2}-W_{2}^{2}=(W_{1}-W_{2})(W_{1}+W_{2})$, and by claim 4 of Properties of the Absolute Value in an Ordered Field, Preliminaries (i) and (iii),

$$|\Phi(\mu')-\Phi(\mu)|=\tfrac12|W_{1}-W_{2}|\,(W_{1}+W_{2})\le\tfrac12W_{\mathbb{T}}(\mu,\mu')\cdot\sqrt{d}.$$

Let $\varepsilon$ be a positive real and put $\delta=\varepsilon/\sqrt{d}$, positive since $d\ge1$. If $W_{\mathbb{T}}(\mu,\mu')<\delta$, then $|\Phi(\mu')-\Phi(\mu)|\le\tfrac{\sqrt{d}}{2}W_{\mathbb{T}}(\mu,\mu')<\tfrac{\varepsilon}{2}<\varepsilon$. As $\mu$ was arbitrary, $\Phi$ is continuous on $\mathcal{P}(\mathbb{T}^{d})$ from $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$ to $\mathbb{R}$ with the absolute-value metric, which is Intrinsic Test Functions on the Torus Wasserstein Space §continuity.

Step 4 (Clause 2, property (b) and the gradient). Let $\mu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ and let $T$ be any optimal map from $\mu$ to $\nu$. By Half the Squared Torus Wasserstein Distance to a Fixed Measure: the One-Sided Bound Along Couplings, Differentiability at an Absolutely Continuous Source, and the Tangent Gradient §differentiable, $\Phi$ is differentiable along couplings at $\mu$ with $\nabla\Phi(\mu)=-v_{T}$, which is the asserted formula for the gradient. By Half the Squared Torus Wasserstein Distance to a Fixed Measure: the One-Sided Bound Along Couplings, Differentiability at an Absolutely Continuous Source, and the Tangent Gradient §tangent the class of $v_{T}$ lies in $T_{\mu}$, which is a linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$ by The Torus Tangent Space: Linearity of the Periodic Calculus, Closed Subspace, and Representation of Bounded Functionals on Gradients §subspace; so the class of $-v_{T}$, its product with the scalar $-1$, lies in $T_{\mu}$. This is Intrinsic Test Functions on the Torus Wasserstein Space §differentiability for $\mathcal{K}=\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$.

Step 5 (Clause 2, property (c)). Let $\mu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, and let $\gamma_{n}\in\Pi(\mu_{n},\mu)$ be such that $(I_{\mathbb{T}}(\gamma_{n}))_{n}$ converges to $0$. By 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 §map choose an optimal map $T$ from $\mu$ to $\nu$ and, for each $n\in\mathbb{N}$, an optimal map $T_{n}$ from $\mu_{n}$ to $\nu$. By Step 4, $\nabla\Phi(\mu)=-v_{T}$ and $\nabla\Phi(\mu_{n})=-v_{T_{n}}$. The terms of the sequence in Intrinsic Test Functions on the Torus Wasserstein Space §gradient-continuity do not depend on the representatives chosen, as recorded there; with the Borel representatives $-v_{T_{n}}$ and $-v_{T}$ and claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ (with $\lambda=-1$), the $n$th term is

$$\int_{\mathbb{R}^{d+d}}\bigl\lVert-v_{T_{n}}(\mathrm{pr}_{1}(w))+v_{T}(\mathrm{pr}_{2}(w))\bigr\rVert^{2}\,\gamma_{n}(dw)=\int_{\mathbb{R}^{d+d}}\bigl\lVert v_{T_{n}}(\mathrm{pr}_{1}(w))-v_{T}(\mathrm{pr}_{2}(w))\bigr\rVert^{2}\,\gamma_{n}(dw),$$

and this sequence converges to $0$ by clause 1, applied to $\mu$, $\mu_{n}$, $T$, $T_{n}$ and $\gamma_{n}$. This is Intrinsic Test Functions on the Torus Wasserstein Space §gradient-continuity.

By Steps 3, 4 and 5, $\Phi$ has the three properties of Intrinsic Test Functions on the Torus Wasserstein Space §test on $\mathcal{K}=\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, which is clause 2.