Proof of McCann's Tangent Inequality on the Torus, and Monotonicity of the Score Along Optimal Maps

Each result cited is universally quantified over the data in its own statement; a result that fixes its own dimension ($n$, $m$, $q$ or $d$) is applied with that dimension equal to the torus dimension $d$ of Optimal Transport on the Flat Torus: Standing Notation §conventions. Integrals against Lebesgue measure are written $\int\cdots d\lambda_{d}$. A Borel set $F$ is $\mu$-full if $\mu(\mathbb{R}^{d}\setminus F)=0$; countably many $\mu$-full sets have a $\mu$-full intersection (claim 4 of Basic Properties of a Measure), and $\mu(E)=\mu(E\cap F)$ for Borel $E$ and $\mu$-full $F$, since $\mu(E)=\mu(E\cap F)+\mu(E\setminus F)$ and $\mu(E\setminus F)\le\mu(\mathbb{R}^{d}\setminus F)=0$ (claims 1 and 2 of that lemma); likewise for the other measures below. To avoid a clash with the potential $\varphi$ below, the function $s\mapsto s\log s$ (value $0$ at $0$) of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm is written $\eta$. We use $\log(ab)=\log a+\log b$ and $\log(a^{-1})=-\log a$ for positive $a,b$, since $\log$ is the inverse of $\exp$ (The Natural Logarithm) and $\exp$ turns sums into products (Basic Properties of the Exponential Function); and if $0<a\le b$ then $\log a\le\log b$, because $\log a-\log b=\log(a/b)\le a/b-1\le0$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log. For $k\in\mathbb{Z}^{d}$ and a Borel $A\subseteq\mathbb{R}^{d}$ we write $A+k=\{a+k:a\in A\}$, which is Borel by claim 1 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$; in particular $Q+k$ is Borel, $Q$ being Borel by The Half-Open Unit Cell Tiles Euclidean Space §cell. Fix a bijection $j\mapsto k_{j}$ from $\mathbb{N}$ onto $\mathbb{Z}^{d}$ (The Integer Lattice Admits an Enumeration by the Natural Numbers §enumeration); by The Half-Open Unit Cell Tiles Euclidean Space §tiling every Borel $A$ is the union of the Borel sets $A\cap(Q+k_{j})$, $j\in\mathbb{N}$.

Proof of clause 1. Let $\mu,\nu$ and $T$ be as in clause 1. Fix a Borel representative $\xi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ of the score $\xi_{\mu}$, so that $\int\lVert\xi\rVert^{2}\,d\mu<\infty$ and $\langle\xi_{\mu},w\rangle_{\mu}=\int\xi\cdot w\,d\mu$ for every Borel $w$ with $\int\lVert w\rVert^{2}\,d\mu<\infty$ (Optimal Transport on the Flat Torus: Standing Notation §fields).

Step 0: densities and the periodic potential. By The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §finite-entropy-set and The Entropy of a Probability Measure on Euclidean Space §entropy, $\mu$ and $\nu$ have densities $\rho$ and $\rho_{1}$ with respect to $\lambda_{d}$ (Borel, real, nonnegative) with $\eta\circ\rho$ and $\eta\circ\rho_{1}$ integrable, $\mathrm{Ent}(\mu)=\int\eta\circ\rho\,d\lambda_{d}$ and $\mathrm{Ent}(\nu)=\int\eta\circ\rho_{1}\,d\lambda_{d}$. Both measures are absolutely continuous by The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §nonnegative, and $\mu(Q)=\nu(Q)=1$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures. Let $P=\{\rho>0\}$ and $P_{1}=\{\rho_{1}>0\}$, and let $\ell,\ell_{1}:\mathbb{R}^{d}\to\mathbb{R}$ equal $\log\rho$ on $P$, respectively $\log\rho_{1}$ on $P_{1}$, and $0$ elsewhere; they are Borel, being the product of $\mathbf{1}_{P}$ with the composite of the continuous $\log$ on $(0,\infty)$ and the positive Borel function $\rho+\mathbf{1}_{\mathbb{R}^{d}\setminus P}$ (likewise for $\ell_{1}$). Since $\ell\rho=\eta\circ\rho$ everywhere (both vanish off $P$), claim 3 of Image Measures, Measures with Densities, and Change of Variables shows that $\ell$ is $\mu$-integrable with $\int\ell\,d\mu=\int\eta\circ\rho\,d\lambda_{d}=\mathrm{Ent}(\mu)$; likewise $\ell_{1}$ is $\nu$-integrable with $\int\ell_{1}\,d\nu=\mathrm{Ent}(\nu)$. As $T$ is an optimal map on the torus, it is Borel with $T_{\#}\mu=\nu$, so by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward

$$\ell_{1}\circ T\ \text{is $\mu$-integrable and}\ \int\ell_{1}\circ T\,d\mu=\mathrm{Ent}(\nu).\qquad(1)$$

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 §potential, applied to $T$ ($\mu$ being absolutely continuous), fix a $\mathbb{Z}^{d}$-periodic $\varphi:\mathbb{R}^{d}\to\mathbb{R}$, Lipschitz with constant $L=\sqrt{d}/2$, such that $\psi(x)=\tfrac12\lVert x\rVert^{2}-\varphi(x)$ is convex on $\mathbb{R}^{d}$, and a Borel $D$ with $\mu(D)=1$ such that $\partial_{\mathbb{R}^{d}}\psi(x)=\{x+v_{T}(x)\}$ for every $x\in D$. By A Lipschitz Map is Uniformly Continuous, $\varphi$ is continuous.

Step 1: the lifted map. Let $\tilde T=\mathrm{id}+v_{T}$; it is Borel, its components being sums of Borel 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; $v_{T}$ is Borel by Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement). By that clause $v_{T}(x)=\varpi(T(x)-x)$, so The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range gives, for every $x$,

$$k(x):=T(x)-\tilde T(x)=(T(x)-x)-\varpi(T(x)-x)\in\mathbb{Z}^{d},\qquad\lVert x-\tilde T(x)\rVert^{2}=\lVert v_{T}(x)\rVert^{2}\le d/4,$$

and $\partial_{\mathbb{R}^{d}}\psi(x)=\{\tilde T(x)\}$ for $x\in D$.

(F) Let $x$ satisfy $T(x)\in Q$. Then $\tilde T(x)-(-k(x))=T(x)\in Q$ with $-k(x)\in\mathbb{Z}^{d}$, so by The Half-Open Unit Cell Tiles Euclidean Space §tiling $-k(x)$ is the only $m\in\mathbb{Z}^{d}$ with $\tilde T(x)-m\in Q$. Hence $\pi(\tilde T(x))=T(x)$ by the definition of $\pi$ in The Half-Open Unit Cell Tiles Euclidean Space §wrap; and if $\tilde T(x)\in Q+k$ for some $k\in\mathbb{Z}^{d}$, that is $\tilde T(x)-k\in Q$, then $k=-k(x)$ and $T(x)=\tilde T(x)-k$.

The set $T^{-1}(Q)$ is Borel and $\mu$-full, since $\mu(T^{-1}(Q))=\nu(Q)=1$. Let $\tilde\nu=\tilde T_{\#}\mu\in\mathcal{P}(\mathbb{R}^{d})$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward).

Step 2: properties of $\tilde\nu$. (a) $\pi_{\#}\tilde\nu=\nu$. The map $\pi$ is Borel (The Half-Open Unit Cell Tiles Euclidean Space §wrap). For Borel $B$, (F) gives $\tilde T^{-1}(\pi^{-1}(B))\cap T^{-1}(Q)=T^{-1}(B)\cap T^{-1}(Q)$, so, $T^{-1}(Q)$ being $\mu$-full, $\tilde\nu(\pi^{-1}(B))=\mu(\tilde T^{-1}(\pi^{-1}(B)))=\mu(T^{-1}(B))=\nu(B)$.

(b) Second moments. Every $x\in Q$ has coordinates in $[0,1)$, so $\lVert x\rVert^{2}\le d$; as $\mu(Q)=1$, the function $\lVert x\rVert^{2}$ agrees $\mu$-almost everywhere with $\mathbf{1}_{Q}(x)\lVert x\rVert^{2}\le d$, and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison gives $\int\lVert x\rVert^{2}\,\mu(dx)\le d$, so $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space). By 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 §pushforward, $\gamma_{0}=(\mathrm{id},\tilde T)_{\#}\mu\in\Pi(\mu,\tilde\nu)$ and $I(\gamma_{0})=\int\lVert x-\tilde T(x)\rVert^{2}\,\mu(dx)\le d/4$, so $\tilde\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by the converse part of 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 §cost-finite.

(c) Domination. Let $u=\rho_{1}\circ\pi$, a nonnegative Borel function. Let $k\in\mathbb{Z}^{d}$ and let $A\subseteq Q+k$ be Borel; we show

$$\tilde\nu(A)\le\nu(A+(-k))=\int\mathbf{1}_{A}u\,d\lambda_{d}\le1.\qquad(2)$$

If $x\in\tilde T^{-1}(A)\cap T^{-1}(Q)$, then $\tilde T(x)\in Q+k$, so $T(x)=\tilde T(x)-k\in A+(-k)$ by (F); hence $\tilde\nu(A)=\mu(\tilde T^{-1}(A)\cap T^{-1}(Q))\le\mu(T^{-1}(A+(-k)))=\nu(A+(-k))\le1$. Let $g=\mathbf{1}_{A+(-k)}\rho_{1}$, Borel and nonnegative, with $\int g\,d\lambda_{d}=\nu(A+(-k))$ because $\rho_{1}$ is a density of $\nu$. For $y\in A$ we have $y-k\in Q$, so $\pi(y)=y-k$ (The Half-Open Unit Cell Tiles Euclidean Space §wrap) and $g(y-k)=\rho_{1}(y-k)=u(y)$; for $y\notin A$ we have $y-k\notin A+(-k)$ and $g(y-k)=0$. Thus $\mathbf{1}_{A}u$ is the function $y\mapsto g(y-k)$, and claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ (with $a=-k$) gives $\int\mathbf{1}_{A}u\,d\lambda_{d}=\int g\,d\lambda_{d}=\nu(A+(-k))$.

(d) Absolute continuity and a density. Let $N$ be Borel with $\lambda_{d}(N)=0$. For each $j$, $\mathbf{1}_{N\cap(Q+k_{j})}u$ vanishes off the null set $N$, so by (2) and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral $\tilde\nu(N\cap(Q+k_{j}))=0$; by countable subadditivity (claim 4 of Basic Properties of a Measure) $\tilde\nu(N)=0$. So $\tilde\nu$ is absolutely continuous, and since $\lambda_{d}$ is $\sigma$-finite (Lebesgue Measure on Euclidean Space is Sigma-Finite §sigma-finite), The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities §existence gives a density $\theta$ of $\tilde\nu$ with respect to $\lambda_{d}$ (Borel, real, nonnegative).

(e) Comparison of densities. Let $B=\{y:\theta(y)>u(y)\}$, a Borel set, and $B_{j}=B\cap(Q+k_{j})$. By (2), $\int\mathbf{1}_{B_{j}}\theta\,d\lambda_{d}=\tilde\nu(B_{j})\le\int\mathbf{1}_{B_{j}}u\,d\lambda_{d}\le1$. The function $w_{j}=\mathbf{1}_{B_{j}}(\theta-u)$ is Borel and nonnegative with $\mathbf{1}_{B_{j}}\theta=\mathbf{1}_{B_{j}}u+w_{j}$, so claim 1 of Linearity and Monotonicity of the Lebesgue Integral gives $\int\mathbf{1}_{B_{j}}u\,d\lambda_{d}+\int w_{j}\,d\lambda_{d}\le\int\mathbf{1}_{B_{j}}u\,d\lambda_{d}$, and, the first term being a real number, $\int w_{j}\,d\lambda_{d}=0$. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing, $w_{j}=0$ $\lambda_{d}$-almost everywhere; as $w_{j}>0$ on $B_{j}$, $\lambda_{d}(B_{j})=0$, hence $\tilde\nu(B_{j})=0$ by (d), and $\tilde\nu(B)=0$ by countable subadditivity. Consequently the Borel set $E=T^{-1}(Q)\setminus\tilde T^{-1}(B)$ is $\mu$-full ($\mu(\tilde T^{-1}(B))=\tilde\nu(B)=0$), and by (F)

$$\theta(\tilde T(x))\le u(\tilde T(x))=\rho_{1}(\pi(\tilde T(x)))=\rho_{1}(T(x))\qquad(x\in E).\qquad(3)$$

Step 3: $\tilde T$ is a Euclidean optimal map from $\mu$ to $\tilde\nu$. Let $\sigma\in\Pi(\mu,\tilde\nu)$. By 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 §modification (with the Borel map $\pi$ in the role of $T$) and (a), $\sigma'=(\mathrm{pr}_{1},\pi\circ\mathrm{pr}_{2})_{\#}\sigma\in\Pi(\mu,\pi_{\#}\tilde\nu)=\Pi(\mu,\nu)$. The pairing $(\mathrm{pr}_{1},\pi\circ\mathrm{pr}_{2})$ has components $\mathrm{pr}_{1}$ and $\pi\circ\mathrm{pr}_{2}$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing), so the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, applied to the Borel function $w\mapsto d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))^{2}$ (The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz), gives for the torus cost

$$I_{\mathbb{T}}(\sigma')=\int d_{\mathbb{T}}\bigl(\mathrm{pr}_{1}(w),\pi(\mathrm{pr}_{2}(w))\bigr)^{2}\,\sigma(dw).$$

For $x,y\in\mathbb{R}^{d}$, $\pi(y)=y-m$ with $m\in\mathbb{Z}^{d}$ (The Half-Open Unit Cell Tiles Euclidean Space §wrap), so $d_{\mathbb{T}}(x,\pi(y))=d_{\mathbb{T}}(x,y)$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §symmetry, and $0\le d_{\mathbb{T}}(x,y)\le\lVert x-y\rVert$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §minimal. By monotonicity of the integral and Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance,

$$W_{\mathbb{T}}(\mu,\nu)^{2}\le I_{\mathbb{T}}(\sigma')\le\int\lVert\mathrm{pr}_{1}(w)-\mathrm{pr}_{2}(w)\rVert^{2}\,\sigma(dw)=I(\sigma).$$

On the other hand $d_{\mathbb{T}}(x,T(x))=\lVert\varpi(T(x)-x)\rVert=\lVert x-\tilde T(x)\rVert$ by The Wrapped Displacement and the Flat Torus Distance §distance, and $(\mathrm{id},T)_{\#}\mu$ is optimal on the torus (Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §map, Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §optimal), so by (b) and the same change of variables for $(\mathrm{id},T)$,

$$I(\gamma_{0})=\int\lVert x-\tilde T(x)\rVert^{2}\,\mu(dx)=\int d_{\mathbb{T}}(x,T(x))^{2}\,\mu(dx)=I_{\mathbb{T}}\bigl((\mathrm{id},T)_{\#}\mu\bigr)=W_{\mathbb{T}}(\mu,\nu)^{2}\le I(\sigma).$$

By Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment, applied to $\mu,\tilde\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, there is $\sigma^{*}\in\Pi(\mu,\tilde\nu)$ with $I(\sigma^{*})=W_{2}(\mu,\tilde\nu)^{2}$ and $I(\sigma^{*})\le I(\gamma_{0})$; with $\sigma=\sigma^{*}$ above, $I(\gamma_{0})=W_{2}(\mu,\tilde\nu)^{2}$. So $\gamma_{0}$ is optimal, and as $\tilde T_{\#}\mu=\tilde\nu$, $\tilde T$ is an optimal map from $\mu$ to $\tilde\nu$.

Step 4: the Jacobian equation. By Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment fix an optimal coupling $\sigma_{1}\in\Pi(\tilde\nu,\mu)$. By Brenier's Theorem: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map §potential, applied to $\sigma_{1}$ with the absolutely continuous $\tilde\nu$ as first measure, fix a Borel $S$ with $\sigma_{1}=(\mathrm{id},S)_{\#}\tilde\nu$, an open convex $G'$ with $\tilde\nu(G')=1$, a convex $\psi':G'\to\mathbb{R}$ and a Borel $D'\subseteq G'$ with $\tilde\nu(D')=1$ and $\partial_{G'}\psi'(y)=\{S(y)\}$ for $y\in D'$. By Brenier's Theorem: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map §map, $S_{\#}\tilde\nu=\mu$, so $S$ is an optimal map from $\tilde\nu$ to $\mu$. The set $\mathbb{R}^{d}$ is open (claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) and convex, with $\mu(\mathbb{R}^{d})=1$, $\psi$ is convex on it, $D\subseteq\mathbb{R}^{d}$ and $\partial_{\mathbb{R}^{d}}\psi(x)=\{\tilde T(x)\}$ for $x\in D$ (Step 1). Apply McCann's Jacobian Equation Along an Optimal Map Between Absolutely Continuous Measures §jacobian to $\mu,\tilde\nu$ (in $\mathcal{P}_{2}(\mathbb{R}^{d})$ by (b), absolutely continuous, with densities $\rho,\theta$), $\tilde T$, $S$, $G=\mathbb{R}^{d}$, $G'$, $\psi$ in the role of that lemma's $\varphi$, $\psi'$ in the role of its $\psi$, $D$ and $D'$. It gives a Borel $X\subseteq D$ with $\mu(X)=1$ such that every $x\in X$ satisfies, among the properties listed in Along an Optimal Map between Absolutely Continuous Measures the Hessians of the Two Convex Potentials are Inverse Matrices §hessians: $\psi$ is twice differentiable at $x$ with first-order coefficient $\tilde T(x)$ and $D^{2}\psi(x)$ is positive definite; and

$$\rho(x)=\theta(\tilde T(x))\,\det D^{2}\psi(x),\qquad\rho(x)>0.$$

Let $\Delta_{X}:\mathbb{R}^{d}\to\mathbb{R}$ be $\mathrm{tr}\,D^{2}\psi(x)$ for $x\in X$ and $0$ elsewhere; it is Borel by The Points of Twice Differentiability of a Convex Function: a Borel Set of Full Measure, and Borel Measurability of the Gradient and Hessian on It §borel (with $U=\mathbb{R}^{d}$, $f=\psi$, $A=X$) and nonnegative by The Points of Twice Differentiability of a Convex Function: a Borel Set of Full Measure, and Borel Measurability of the Gradient and Hessian on It §nonnegative.

Step 5: the pointwise inequality. Let $X'=X\cap E$, a $\mu$-full Borel set, and $x\in X'$. By Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$ §positive, $\det D^{2}\psi(x)>0$, so $\theta(\tilde T(x))=\rho(x)(\det D^{2}\psi(x))^{-1}>0$, and by (3) $\rho_{1}(T(x))\ge\theta(\tilde T(x))>0$; thus $x\in P$, $T(x)\in P_{1}$, and by monotonicity of $\log$, $\log\theta(\tilde T(x))\le\ell_{1}(T(x))$. By Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$ §log-det, $\log\det D^{2}\psi(x)\le\mathrm{tr}\,D^{2}\psi(x)-d=\Delta_{X}(x)-d$. Taking logarithms in the Jacobian equation,

$$\ell(x)=\log\theta(\tilde T(x))+\log\det D^{2}\psi(x)\le\ell_{1}(T(x))+\Delta_{X}(x)-d\qquad(x\in X').\qquad(4)$$

Step 6: the torus score. We show that $\Delta_{X}$ is $\mu$-integrable and

$$\int\Delta_{X}\,d\mu\le d-\langle\xi_{\mu},v_{T}\rangle_{\mu}.\qquad(5)$$

(i) Second-order expansion of $\varphi$ on $X$. Let $x\in X$, $p=\tilde T(x)$ and $B=D^{2}\psi(x)$. Since $\tfrac12\lVert x+h\rVert^{2}-\tfrac12\lVert x\rVert^{2}=x\cdot h+\tfrac12h\cdot h$ and $\varphi=\tfrac12\lVert\cdot\rVert^{2}-\psi$, for every $h\in\mathbb{R}^{d}$

$$\varphi(x+h)-\varphi(x)-(x-p)\cdot h-\tfrac12h\cdot\bigl((I_{d}-B)h\bigr)=-\Bigl(\psi(x+h)-\psi(x)-p\cdot h-\tfrac12h\cdot(Bh)\Bigr),$$

and $I_{d}-B\in\mathcal{S}(d)$. So, by Twice Differentiability at a Point §twice-differentiable, $\varphi$ is twice differentiable at $x$ with first-order coefficient $x-\tilde T(x)=-v_{T}(x)$ and Hessian $I_{d}-D^{2}\psi(x)$.

(ii) Periodic mollification. Let $\kappa$ be a mollifier kernel of radius $1$ on $\mathbb{R}^{d}$ (claim 2 of Existence of Mollifier Kernels of Every Radius); for $m\in\mathbb{N}$ let $\kappa_{1/m}(y)=m^{d}\kappa(my)$, a mollifier kernel of radius $1/m$ by Rescaling a Mollifier Kernel (with $\varepsilon=m^{-1}$), and $\varphi_{m}=\varphi*\kappa_{1/m}$ (Convolution of a Continuous Function with a Compactly Supported Continuous Kernel with $\Omega=\mathbb{R}^{d}$, whose $\Omega^{1/m}$ is $\mathbb{R}^{d}$). By claim 2 of Convolution with a $C^k$ Kernel is of Class $C^k$, $\varphi_{m}$ is smooth on $\mathbb{R}^{d}$. For $k\in\mathbb{Z}^{d}$ the integrands $y\mapsto\varphi(x+k-y)\kappa_{1/m}(y)$ and $y\mapsto\varphi(x-y)\kappa_{1/m}(y)$ defining $\varphi_{m}(x+k)$ and $\varphi_{m}(x)$ coincide, $\varphi$ being periodic; so $\varphi_{m}\in C^{\infty}_{\mathrm{per}}$ (Lattice-Periodic Functions and the Periodic Function Classes §classes).

(iii) A uniform gradient bound. For $x,x'\in\mathbb{R}^{d}$, claim 2 of Linearity and Monotonicity of the Lebesgue Integral, the bound $|\varphi(x-y)-\varphi(x'-y)|\le L\lVert x-x'\rVert$, and the nonnegativity and unit mass of $\kappa_{1/m}$ (Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$) give

$$|\varphi_{m}(x)-\varphi_{m}(x')|\le\int|\varphi(x-y)-\varphi(x'-y)|\,\kappa_{1/m}(y)\,\lambda_{d}(dy)\le L\lVert x-x'\rVert .$$

So every difference quotient $t^{-1}(\varphi_{m}(x+te_{i})-\varphi_{m}(x))$, $t\ne0$, lies in $[-L,L]$, and so does its limit $\partial_{i}\varphi_{m}(x)$ as $t\to0$; hence $\lVert\nabla\varphi_{m}(x)\rVert^{2}=\sum_{i}(\partial_{i}\varphi_{m}(x))^{2}\le dL^{2}$ and $\lVert\nabla\varphi_{m}(x)\rVert\le\sqrt{d}\,L=d/2$ for all $m$ and $x$.

(iv) One-sided Hessian bound. The continuous function $-\varphi$ is semiconvex on $\mathbb{R}^{d}$ with constant $1$, since $-\varphi+\tfrac12\lVert\cdot\rVert^{2}=\psi$ is convex. By Mollification Preserves Semiconvexity (with $\Omega=\mathbb{R}^{d}$ and the kernel $\kappa_{1/m}$ of radius $1/m$), $(-\varphi)*\kappa_{1/m}$ is semiconvex on $\mathbb{R}^{d}$ with constant $1$, and $(-\varphi)*\kappa_{1/m}=-\varphi_{m}$ by linearity of the integral. So $g_{m}=-\varphi_{m}+\tfrac12\lVert\cdot\rVert^{2}$ is convex on $\mathbb{R}^{d}$; it is of class $C^{2}$ (Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set), with $\partial_{j}\partial_{i}g_{m}=\delta_{ij}-\partial_{j}\partial_{i}\varphi_{m}$, that is $D^{2}g_{m}(x)=I_{d}-D^{2}\varphi_{m}(x)$. By A Convex Function of Class $C^2$ has Positive Semidefinite Hessian, $0\preceq D^{2}g_{m}(x)$; testing The Positive Semidefinite Ordering on Symmetric Matrices with $z=e_{i}$ gives $1-\partial_{i}\partial_{i}\varphi_{m}(x)\ge0$, and summing over $i$ (The Laplacian of a Twice Continuously Differentiable Function §laplacian)

$$f_{m}(x):=d-\Delta\varphi_{m}(x)\ge0\qquad(x\in\mathbb{R}^{d},\ m\in\mathbb{N}).\qquad(6)$$

(v) Convergence on $X$. Let $x\in X$. By (i) and Mollification at a Point of Twice Differentiability: Convergence of the Mollified Gradient and Hessian, and a Hessian Bound for Lipschitz Functions §convergence (with $f=\varphi$, continuous, the kernel $\kappa$ of radius $1$ and $\varepsilon_{m}=m^{-1}\to0$), $\nabla\varphi_{m}(x)\to-v_{T}(x)$ and $D^{2}\varphi_{m}(x)\to I_{d}-D^{2}\psi(x)$ in $\mathcal{S}(d)$. By Limits and Bounded Sequences of Symmetric Real Matrices §quadratic-form with $z=e_{i}$ the diagonal entries converge, so $\Delta\varphi_{m}(x)=\sum_{i}\partial_{i}\partial_{i}\varphi_{m}(x)\to\sum_{i}\bigl(1-(D^{2}\psi(x))_{ii}\bigr)=d-\Delta_{X}(x)$ (Trace of a Real Square Matrix); that is, $f_{m}(x)\to\Delta_{X}(x)$.

(vi) The torus integration by parts. Since $\varphi_{m}\in C^{\infty}_{\mathrm{per}}$, the defining identity of the score, Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §score, applies to $f=\varphi_{m}$, where $\nabla\varphi_{m}$ and the bounded continuous $\Delta\varphi_{m}$ are as in Optimal Transport on the Flat Torus: Standing Notation §calculus:

$$\int\xi\cdot\nabla\varphi_{m}\,d\mu=\langle\xi_{\mu},\nabla\varphi_{m}\rangle_{\mu}=-\int\Delta\varphi_{m}\,d\mu,\qquad\text{so}\qquad\int f_{m}\,d\mu=d+\int\xi\cdot\nabla\varphi_{m}\,d\mu\qquad(7)$$

by claim 2 of Linearity and Monotonicity of the Lebesgue Integral.

(vii) Limit of the pairings. The function $\lVert\xi\rVert$ is Borel (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions) and square-integrable, and so $\mu$-integrable by Hoelder's Inequality, for Two and for Finitely Many Factors §holder (exponents $2,2$, the constant $1$ being square-integrable for the probability measure $\mu$). By Cauchy-Schwarz Inequality for the Euclidean Dot Product and (iii), $|\xi\cdot\nabla\varphi_{m}|\le\tfrac{d}{2}\lVert\xi\rVert$ everywhere, and by (v) $\xi(x)\cdot\nabla\varphi_{m}(x)\to-\xi(x)\cdot v_{T}(x)$ for $x$ in the $\mu$-full set $X$ (coordinatewise convergence in the finite sum defining the dot product). By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §dominated,

$$\int\xi\cdot\nabla\varphi_{m}\,d\mu\to-\int\xi\cdot v_{T}\,d\mu=-\langle\xi_{\mu},v_{T}\rangle_{\mu}.$$

(viii) Fatou. The functions $f_{m}$ are continuous, hence Borel, and nonnegative by (6). For $x\in X$ the sequence $(f_{m}(x))$ converges to $\Delta_{X}(x)$ by (v), so its lower limit is $\Delta_{X}(x)$; for $x\notin X$, $\Delta_{X}(x)=0$ is at most the lower limit. By monotonicity of the integral, Fatou's Lemma, (7) and (vii),

$$\int\Delta_{X}\,d\mu\le\int\Bigl(\liminf_{m}f_{m}\Bigr)d\mu\le\liminf_{m}\int f_{m}\,d\mu=\lim_{m}\Bigl(d+\int\xi\cdot\nabla\varphi_{m}\,d\mu\Bigr)=d-\langle\xi_{\mu},v_{T}\rangle_{\mu}.$$

In particular $\int\Delta_{X}\,d\mu<\infty$, so the nonnegative Borel $\Delta_{X}$ is $\mu$-integrable, and (5) holds.

Step 7: integration. The function $h=\ell_{1}\circ T+\Delta_{X}-d-\ell$ is $\mu$-integrable by (1), Step 0, Step 6 and claim 2 of Linearity and Monotonicity of the Lebesgue Integral (constants being integrable for the probability measure $\mu$). By (4), $h\ge0$ on the $\mu$-full set $X'$, so $\mathbf{1}_{X'}h$ is nonnegative everywhere and equals $h$ $\mu$-almost everywhere; by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison and monotonicity, $\int h\,d\mu=\int\mathbf{1}_{X'}h\,d\mu\ge0$. By linearity, (1), Step 0 and (5),

$$\mathrm{Ent}(\mu)\le\mathrm{Ent}(\nu)+\int\Delta_{X}\,d\mu-d\le\mathrm{Ent}(\nu)-\langle\xi_{\mu},v_{T}\rangle_{\mu},$$

which is clause 1.

Proof of clause 2. Let $\mu,\nu,T,S$ be as in clause 2. Clause 1 applied to $\mu,\nu,T$ ($\mu,\nu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$, $\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$) gives $\mathrm{Ent}(\mu)+\langle\xi_{\mu},v_{T}\rangle_{\mu}\le\mathrm{Ent}(\nu)$, and clause 1 applied to $\nu,\mu,S$ ($\nu,\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$, $\nu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$, $S$ an optimal map from $\nu$ to $\mu$) gives $\mathrm{Ent}(\nu)+\langle\xi_{\nu},v_{S}\rangle_{\nu}\le\mathrm{Ent}(\mu)$. Adding these inequalities between real numbers and subtracting the real number $\mathrm{Ent}(\mu)+\mathrm{Ent}(\nu)$ from both sides gives $\langle\xi_{\mu},v_{T}\rangle_{\mu}+\langle\xi_{\nu},v_{S}\rangle_{\nu}\le0$.