Proof of Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information

Each result cited below is universally quantified over the data in its own statement and is applied to the data named where it is used; The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails, Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity and The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral are read with $q=d$, and The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder with $m=d$.

Conventions. The number $s$ with $0<s\le\tfrac12$ is fixed. Integrals $\int f(y)\,dy$ are over $\mathbb{R}^{d}$ with respect to $\lambda_{d}$. Let $c_{s}>0$ be the constant of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails, so that $g_{s}(z)=c_{s}\exp(-\lVert z\rVert^{2}/(2s))$. As recorded there, $g_{s}$ is the Gaussian smoothing weight of claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder with $\eta=s$; by that claim $g_{s}$ is Borel, $0<g_{s}(z)$ and $g_{s}(-z)=g_{s}(z)$ for every $z$, and $\int g_{s}(z-a)\,dz=1$ for every $a\in\mathbb{R}^{d}$. For $\kappa\in\mathcal{P}(\mathbb{R}^{d})$ let $g_{s}*\kappa$ be the Gaussian smoothing of $\kappa$ at scale $s$, nonnegative, Borel and of integral $1$ by Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity §duality, and let $\kappa_{s}\in\mathcal{P}(\mathbb{R}^{d})$ be the measure with density $g_{s}*\kappa$ with respect to $\lambda_{d}$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables), exactly as in The Heat Semigroup on the Probability Measures on the Torus; thus $S_{s}\kappa=\pi_{\#}\kappa_{s}$ for $\kappa\in\mathcal{P}(\mathbb{T}^{d})$ by The Heat Semigroup on the Probability Measures on the Torus §heat. Densities are those of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities, as in The Entropy of a Probability Measure on Euclidean Space §entropy, and $\phi$ is the function of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm ($\phi(0)=0$, $\phi(t)=t\log t$ for $t>0$) used there.

The result Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability is stated in the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose dimension is the $d$ fixed by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data; that $d$ is an arbitrary natural number with $1\le d$, the result holding for each such choice, and we take it to be the dimension $d$ of the torus. By Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions, the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ and the distance $W_{2}$ named there are those of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space and The Quadratic Wasserstein Distance on Euclidean Space §distance, which are also those of The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions; and for $\kappa\in\mathcal{P}_{2}(\mathbb{R}^{d})$ in place of its $\mu$, its $\rho_{s}$ and $\mu_{s}$ are $g_{s}*\kappa$ and $\kappa_{s}$. So Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability §density gives:

$$\kappa\in\mathcal{P}_{2}(\mathbb{R}^{d})\ \Longrightarrow\ \kappa_{s}\in\mathcal{P}_{2}(\mathbb{R}^{d})\ \text{ and }\ W_{2}(\kappa_{s},\kappa)^{2}\le d\,s. \qquad(\mathrm{G})$$

Step 0 (Preliminaries).

(a) Measures on the torus. Let $\kappa\in\mathcal{P}(\mathbb{T}^{d})$. As $\kappa(Q)=1$ (Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures), claim 3 of Basic Properties of a Measure gives $\kappa(\mathbb{R}^{d}\setminus Q)=0$, and for Borel $A$ finite additivity and monotonicity (claims 1 and 2 there), applied to $A\cap Q$ and $A\setminus Q\subseteq\mathbb{R}^{d}\setminus Q$, give $\kappa(A)=\kappa(A\cap Q)$. The wrapping map $\pi$ is Borel and $\pi(x)=x$ for $x\in Q$ (The Half-Open Unit Cell Tiles Euclidean Space §wrap), so $\pi^{-1}(A)\cap Q=A\cap Q$ and $\pi_{\#}\kappa(A)=\kappa(\pi^{-1}(A)\cap Q)=\kappa(A\cap Q)=\kappa(A)$; that is, $\pi_{\#}\kappa=\kappa$. Also $\kappa\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §inclusion. Finally, for $x\in Q$ one has $0\le x_{i}<1$, so $x_{i}^{2}\le1$, for each $i\in[d]$, whence $\lVert x\rVert^{2}=\sum_{i=1}^{d}x_{i}^{2}\le d$ (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) and $\lVert x\rVert\le\sqrt{d}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field).

(b) Iterated integrals. Let $\iota:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d+d}$ be the concatenation map of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs. If $\hat F:\mathbb{R}^{d+d}\to[0,\infty]$ is Borel, then $(x,y)\mapsto\hat F(\iota(x,y))$ is measurable for $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{B}(\mathbb{R}^{d})$, since $\iota$ is measurable for that $\sigma$-algebra and $\mathcal{B}(\mathbb{R}^{d+d})$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product and a composite of measurable maps is measurable; and $\hat F(\iota(x,y))$ is obtained by substituting $x,y$ for $\mathrm{pr}_{1}(z),\mathrm{pr}_{2}(z)$ in any formula for $\hat F(z)$, by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections. Lebesgue measure is $\sigma$-finite by Lebesgue Measure on Euclidean Space is Sigma-Finite §sigma-finite, and every member of $\mathcal{P}(\mathbb{R}^{d})$ is finite, hence $\sigma$-finite. So the Tonelli clause of Tonelli and Fubini Theorems applies to such integrands for any two of these measures: the iterated integrals exist in either order, are equal, and the partial integrals are measurable functions of the remaining variable. Every integrand to which this is applied below has this form, $\hat F$ being built from $\mathrm{pr}_{1}$, $\mathrm{pr}_{2}$, $\pi$, $g_{s}$, continuous functions, Borel densities and indicators of Borel sets by composition, sums and products, and hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps.

(c) The integer part. For $y\in\mathbb{R}^{d}$ put $n(y)=y-\pi(y)$. By The Half-Open Unit Cell Tiles Euclidean Space §wrap, $n(y)$ is the unique $m\in\mathbb{Z}^{d}$ with $y-m\in Q$ provided by The Half-Open Unit Cell Tiles Euclidean Space §tiling; $n$ is Borel, being the composite of the Borel pairing $(\mathrm{id},\pi)$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing) with the continuous map $z\mapsto\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)$; and $n(y+k)=y+k-\pi(y)=n(y)+k$ for $k\in\mathbb{Z}^{d}$, as $\pi(y+k)=\pi(y)$ by the same clause. We record three facts.

(c1) For $k\in\mathbb{Z}^{d}$: $n(y)=k$ if and only if $y-k\in Q$. Indeed, if $n(y)=k$ then $y-k=\pi(y)\in Q$; conversely, if $y-k\in Q$ then $k$ is the unique $m\in\mathbb{Z}^{d}$ with $y-m\in Q$ (The Half-Open Unit Cell Tiles Euclidean Space §tiling), which is $n(y)$. Hence $\{y:n(y)=k\}=Q+k$, a Borel set with $\lambda_{d}(Q+k)=\lambda_{d}(Q)=1$ by claim 1 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ and The Half-Open Unit Cell Tiles Euclidean Space §cell.

(c2) For $z\in\mathbb{R}^{d}$ and Borel $B\subseteq Q$:

$$\int\mathbf{1}_{B}\bigl(z-n(y)\bigr)\,dy=\mathbf{1}_{B}\bigl(\pi(z)\bigr).$$

Indeed, $n(y)\in\mathbb{Z}^{d}$, so by (c1) applied to $z$ and $k=n(y)$, $z-n(y)\in Q$ holds exactly when $n(y)=n(z)$, and then $z-n(y)=\pi(z)$. As $B\subseteq Q$, the set $\{y:z-n(y)\in B\}$ is $\{y:n(y)=n(z)\}=Q+n(z)$ if $\pi(z)\in B$ and is empty otherwise; its indicator is the integrand, whose integral is its $\lambda_{d}$-measure (Measure Spaces and the Lebesgue Integral: Standing Notation §integral), namely $1$ or $0$ by (c1).

(c3) For $z\in\mathbb{R}^{d}$: $\int\mathbf{1}_{Q}(z+n(y))\,dy=1$. Indeed $-n(y)\in\mathbb{Z}^{d}$, so by (c1) $z+n(y)=z-(-n(y))\in Q$ exactly when $-n(y)=n(z)$, that is, $n(y)=-n(z)$; the set of such $y$ is $Q-n(z)$, of measure $1$ by (c1).

(d) Representation of $\kappa_{s}$. Let $\kappa\in\mathcal{P}(\mathbb{R}^{d})$ and let $h:\mathbb{R}^{d}\to[0,\infty)$ be Borel. Then $x\mapsto\int h(x+z)g_{s}(z)\,dz$ is Borel and

$$\int h\,d\kappa_{s}=\int\Bigl(\int h(x+z)\,g_{s}(z)\,dz\Bigr)\kappa(dx). \qquad(\mathrm{R})$$

By claim 3 of Image Measures, Measures with Densities, and Change of Variables and the definition of $g_{s}*\kappa$, $\int h\,d\kappa_{s}=\int h(y)\bigl(\int g_{s}(y-x)\,\kappa(dx)\bigr)dy=\int\bigl(\int h(y)g_{s}(y-x)\,\kappa(dx)\bigr)dy$, the constant $h(y)$ being taken inside by claim 1 of Linearity and Monotonicity of the Lebesgue Integral. By (b) with the measures $\lambda_{d}$ and $\kappa$ this equals $\int\bigl(\int h(y)g_{s}(y-x)\,dy\bigr)\kappa(dx)$, and the inner integral equals $\int h(x+z)g_{s}(z)\,dz$ by claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ with $a=x$, applied to $y\mapsto h(y)g_{s}(y-x)$. Measurability of the partial integral is part of (b).

Step 1 (Clause 1). By Step 0(a), $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi_{\#}\mu=\mu$; by (G), $\mu_{s}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $W_{2}(\mu_{s},\mu)^{2}\le ds$. The final assertion of The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §wrap gives

$$W_{\mathbb{T}}(S_{s}\mu,\mu)=W_{\mathbb{T}}(\pi_{\#}\mu_{s},\pi_{\#}\mu)\le W_{2}(\mu_{s},\mu),$$

both sides being nonnegative, so $W_{\mathbb{T}}(S_{s}\mu,\mu)^{2}\le W_{2}(\mu_{s},\mu)^{2}\le ds$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Step 2 (Clause 2). By The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §optimal choose an optimal coupling $\gamma\in\Pi(\mu,\nu)$, so that $I_{\mathbb{T}}(\gamma)=W_{\mathbb{T}}(\mu,\nu)^{2}$ (Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §optimal).

(2a) Unwrapping the second marginal. Let $T:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ be $T(w)=\mathrm{pr}_{1}(w)+\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))$, with the wrapped displacement $\varpi$, which is Borel by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz; $T$ is Borel as a composite of Borel maps with continuous vector operations (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps). Put $\tilde\nu=T_{\#}\gamma\in\mathcal{P}(\mathbb{R}^{d})$ and $\tilde\gamma=(\mathrm{pr}_{1},T)_{\#}\gamma\in\mathcal{P}(\mathbb{R}^{d+d})$, the pairing being Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing. Since $\mathrm{pr}_{1}\circ(\mathrm{pr}_{1},T)=\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}\circ(\mathrm{pr}_{1},T)=T$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections) and a push-forward of a push-forward is the push-forward by the composite (as $(S\circ S')^{-1}(B)=S'^{-1}(S^{-1}(B))$), $\tilde\gamma$ has marginals $\mathrm{pr}_{1\#}\gamma=\mu$ and $T_{\#}\gamma=\tilde\nu$, so $\tilde\gamma\in\Pi(\mu,\tilde\nu)$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling). By the change of variables of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward and The Wrapped Displacement and the Flat Torus Distance §distance,

$$I(\tilde\gamma)=\int\lVert\mathrm{pr}_{1}(w)-T(w)\rVert^{2}\,\gamma(dw)=\int\bigl\lVert\varpi\bigl(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w)\bigr)\bigr\rVert^{2}\,\gamma(dw)=\int d_{\mathbb{T}}\bigl(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w)\bigr)^{2}\,\gamma(dw)=I_{\mathbb{T}}(\gamma)=W_{\mathbb{T}}(\mu,\nu)^{2},$$

using $\lVert-a\rVert=\lVert a\rVert$ (claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) and Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost. As $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ (Step 0(a)) and $I(\tilde\gamma)<\infty$, the second assertion 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 gives $\tilde\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Finally, for every $w$, with $v=\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w)$,

$$T(w)-\mathrm{pr}_{2}(w)=-\bigl(v-\varpi(v)\bigr)\in\mathbb{Z}^{d},$$

because $v-\varpi(v)\in\mathbb{Z}^{d}$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range and the negative of an integer vector is an integer vector (Lattice-Periodic Functions and the Periodic Function Classes §lattice).

(2b) Wrapping back: $\pi_{\#}\tilde\nu_{s}=S_{s}\nu$. Let $B\in\mathcal{B}(\mathbb{R}^{d})$, put $\varphi_{B}=\mathbf{1}_{B}\circ\pi$, a Borel function with values in $\{0,1\}$, and let $h_{B}=g_{s}*\varphi_{B}$ be as in Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity §smoothing; by Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity §duality (with $M=1$) $h_{B}$ is Borel with $|h_{B}|\le1$. It is $\mathbb{Z}^{d}$-periodic: for $x\in\mathbb{R}^{d}$ and $k\in\mathbb{Z}^{d}$, apply claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ with $a=k$ to the nonnegative Borel function $f(y)=g_{s}(x+k-y)\varphi_{B}(y)$; since $\varphi_{B}(y+k)=\mathbf{1}_{B}(\pi(y+k))=\varphi_{B}(y)$ (The Half-Open Unit Cell Tiles Euclidean Space §wrap), $f(y+k)=g_{s}(x-y)\varphi_{B}(y)$, so

$$h_{B}(x+k)=\int f(y)\,dy=\int f(y+k)\,dy=\int g_{s}(x-y)\varphi_{B}(y)\,dy=h_{B}(x).$$

For every $\kappa\in\mathcal{P}(\mathbb{R}^{d})$, claim 3 of Image Measures, Measures with Densities, and Change of Variables and Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity §duality give

$$\pi_{\#}\kappa_{s}(B)=\kappa_{s}\bigl(\pi^{-1}(B)\bigr)=\int\varphi_{B}\,(g_{s}*\kappa)\,d\lambda_{d}=\int h_{B}\,d\kappa. \qquad(\mathrm{W})$$

By (2a), $h_{B}(T(w))=h_{B}(\mathrm{pr}_{2}(w))$ for every $w$, $T(w)$ and $\mathrm{pr}_{2}(w)$ differing by an integer vector. Hence, using (W) twice and the change of variables of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward for the bounded Borel function $h_{B}$,

$$\pi_{\#}\tilde\nu_{s}(B)=\int h_{B}\,d\tilde\nu=\int h_{B}\circ T\,d\gamma=\int h_{B}\circ\mathrm{pr}_{2}\,d\gamma=\int h_{B}\,d\nu=\pi_{\#}\nu_{s}(B)=S_{s}\nu(B).$$

As $B$ was arbitrary, $\pi_{\#}\tilde\nu_{s}=S_{s}\nu$.

(2c) Smoothing with a common Gaussian increment. Let $N_{s}\in\mathcal{P}(\mathbb{R}^{d})$ be the measure with density $g_{s}$ with respect to $\lambda_{d}$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables; $\int g_{s}\,dz=1$). Let $\Gamma_{0}=\tilde\gamma\boxtimes N_{s}\in\mathcal{P}(\mathbb{R}^{(d+d)+d})$ be the product measure of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, read with $q=d+d$ and $p=d$, with concatenation $\iota'$ and projections $P_{1}=\mathrm{pr}^{d+d,d}_{1}$, $P_{2}=\mathrm{pr}^{d+d,d}_{2}$. For Borel $\hat F:\mathbb{R}^{(d+d)+d}\to[0,\infty]$, that clause, the Tonelli clause of Tonelli and Fubini Theorems and claim 3 of Image Measures, Measures with Densities, and Change of Variables give

$$\int\hat F\,d\Gamma_{0}=\int\Bigl(\int\hat F\bigl(\iota'(w,z)\bigr)\,g_{s}(z)\,dz\Bigr)\tilde\gamma(dw). \qquad(\mathrm{P})$$

Let $F:\mathbb{R}^{(d+d)+d}\to\mathbb{R}^{d+d}$ send $\zeta$ to $\iota\bigl(\mathrm{pr}_{1}(P_{1}\zeta)+P_{2}\zeta,\ \mathrm{pr}_{2}(P_{1}\zeta)+P_{2}\zeta\bigr)$, a Borel map by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, and put $\Gamma=F_{\#}\Gamma_{0}\in\mathcal{P}(\mathbb{R}^{d+d})$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, $\mathrm{pr}_{1}(F(\iota'(w,z)))=\mathrm{pr}_{1}(w)+z$ and $\mathrm{pr}_{2}(F(\iota'(w,z)))=\mathrm{pr}_{2}(w)+z$. For $A\in\mathcal{B}(\mathbb{R}^{d})$, the change of variables, (P), the change of variables for $\mathrm{pr}_{1\#}\tilde\gamma=\mu$ applied to the Borel function $x\mapsto\int\mathbf{1}_{A}(x+z)g_{s}(z)\,dz$ of Step 0(d), and (R) with $h=\mathbf{1}_{A}$ give

$$\Gamma\bigl(\mathrm{pr}_{1}^{-1}(A)\bigr)=\int\Bigl(\int\mathbf{1}_{A}\bigl(\mathrm{pr}_{1}(w)+z\bigr)g_{s}(z)\,dz\Bigr)\tilde\gamma(dw)=\int\Bigl(\int\mathbf{1}_{A}(x+z)g_{s}(z)\,dz\Bigr)\mu(dx)=\mu_{s}(A),$$

and in the same way, with $\mathrm{pr}_{2}$ and $\mathrm{pr}_{2\#}\tilde\gamma=\tilde\nu$, $\Gamma(\mathrm{pr}_{2}^{-1}(A))=\tilde\nu_{s}(A)$. So $\Gamma\in\Pi(\mu_{s},\tilde\nu_{s})$. Its cost is, by the change of variables and (P),

$$I(\Gamma)=\int\Bigl(\int\bigl\lVert(\mathrm{pr}_{1}(w)+z)-(\mathrm{pr}_{2}(w)+z)\bigr\rVert^{2}g_{s}(z)\,dz\Bigr)\tilde\gamma(dw)=\int\lVert\mathrm{pr}_{1}(w)-\mathrm{pr}_{2}(w)\rVert^{2}\,\tilde\gamma(dw)=I(\tilde\gamma),$$

the inner integrand being the constant $\lVert\mathrm{pr}_{1}(w)-\mathrm{pr}_{2}(w)\rVert^{2}$ times $g_{s}$, of integral $1$ (claim 1 of Linearity and Monotonicity of the Lebesgue Integral).

(2d) Conclusion. By (G) applied to $\mu$ and to $\tilde\nu$, both in $\mathcal{P}_{2}(\mathbb{R}^{d})$, we have $\mu_{s},\tilde\nu_{s}\in\mathcal{P}_{2}(\mathbb{R}^{d})$, so The Quadratic Wasserstein Distance on Euclidean Space §distance gives $W_{2}(\mu_{s},\tilde\nu_{s})^{2}\le I(\Gamma)=I(\tilde\gamma)=W_{\mathbb{T}}(\mu,\nu)^{2}$, hence $W_{2}(\mu_{s},\tilde\nu_{s})\le W_{\mathbb{T}}(\mu,\nu)$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By (2b) and the final assertion of The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §wrap,

$$W_{\mathbb{T}}(S_{s}\mu,S_{s}\nu)=W_{\mathbb{T}}(\pi_{\#}\mu_{s},\pi_{\#}\tilde\nu_{s})\le W_{2}(\mu_{s},\tilde\nu_{s})\le W_{\mathbb{T}}(\mu,\nu).$$

Step 3 (A periodised Gaussian kernel). Let $h_{0}=g_{s}$ and, for $i,j\in[d]$, $h_{i}=\partial_{i}g_{s}$ and $h_{ij}=\partial_{j}\partial_{i}g_{s}$; we call these $1+d+d^{2}$ functions the kernels. By The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §derivatives, $g_{s}$ is smooth, hence of class $C^{3}$ (Smooth Map on a Euclidean Open Set); by clause 2 of C^k Maps on a Euclidean Open Set each $h_{i}$ is of class $C^{2}$ and each $h_{ij}$ exists and is of class $C^{1}$, so every kernel is continuous (claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), hence Borel, and $\partial_{i}h_{0}=h_{i}$, $\partial_{j}h_{i}=h_{ij}$ everywhere. By the same clause of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails, $h_{i}(u)=-s^{-1}u_{i}\,g_{s}(u)$ and $h_{ij}(u)=(s^{-2}u_{i}u_{j}-s^{-1}\delta_{ij})\,g_{s}(u)$.

(3a) A Gaussian bound. Let $e(y)=\exp(-\lVert y\rVert^{2}/2)$; this is the function $\psi_{1}$ of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder ($\eta=1$), so by claim 1 there it is Borel and positive with $\int e\,dy<\infty$, that is, integrable. Put $K_{s}=s^{-1}+s^{-2}$, so $K_{s}\ge1$ as $s^{-1}\ge2$, and for real $R\ge0$ put $a_{R}=R+\sqrt{d}$ and $M_{R}=5K_{s}c_{s}\exp(\tfrac32a_{R}^{2})$. We claim: for every kernel $h$, every $v$ with $\lVert v\rVert\le R$ and every $y$,

$$\bigl|h\bigl(v+n(y)\bigr)\bigr|\le M_{R}\,e(y). \qquad(\mathrm{D})$$

Put $u=v+n(y)$. First, $|h(u)|\le K_{s}(1+\lVert u\rVert^{2})g_{s}(u)$: for $h_{0}$ because $K_{s}\ge1$; for $h_{i}$ because $|u_{i}|\le\lVert u\rVert\le1+\lVert u\rVert^{2}$ (claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, and $0\le(1-\lVert u\rVert)^{2}$) and $s^{-1}\le K_{s}$; for $h_{ij}$ because $|s^{-2}u_{i}u_{j}-s^{-1}\delta_{ij}|\le s^{-2}\lVert u\rVert^{2}+s^{-1}\le K_{s}(1+\lVert u\rVert^{2})$. Next let $t=\lVert u\rVert^{2}/(8s)\ge0$. By claim 4 of Basic Properties of the Exponential Function, $\exp(t)\ge1+t$, so $1\le\exp(t)$ and $\lVert u\rVert^{2}=8st\le8s\exp(t)$, whence $1+\lVert u\rVert^{2}\le(1+8s)\exp(t)\le5\exp(t)$. By claims 1 and 2 there, $g_{s}(u)=c_{s}\exp(-t)\exp(-3t)$ and $\exp(-t)=\exp(t)^{-1}$, so

$$|h(u)|\le5K_{s}c_{s}\exp\Bigl(-\frac{3\lVert u\rVert^{2}}{8s}\Bigr)\le5K_{s}c_{s}\exp\Bigl(-\frac{3\lVert u\rVert^{2}}{4}\Bigr),$$

the last step because $\frac{3}{8s}\ge\frac34$ and $\exp$ is increasing (claim 4 there). Now $y=n(y)+\pi(y)=u-v+\pi(y)$ with $\lVert\pi(y)\rVert\le\sqrt{d}$ (Step 0(a), as $\pi(y)\in Q$), so $\lVert y\rVert\le\lVert u\rVert+a_{R}$ by the triangle inequality (claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$). Since $2\lVert u\rVert a_{R}\le\tfrac12\lVert u\rVert^{2}+2a_{R}^{2}$ (expand $0\le(\tfrac{1}{\sqrt2}\lVert u\rVert-\sqrt2a_{R})^{2}$) and squaring is monotone on nonnegative numbers (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), $\lVert y\rVert^{2}\le(\lVert u\rVert+a_{R})^{2}\le\tfrac32\lVert u\rVert^{2}+3a_{R}^{2}$, that is, $\tfrac34\lVert u\rVert^{2}\ge\tfrac12\lVert y\rVert^{2}-\tfrac32a_{R}^{2}$. Using once more that $\exp$ is increasing and multiplicative, $\exp(-\tfrac34\lVert u\rVert^{2})\le\exp(\tfrac32a_{R}^{2})\,e(y)$, which proves (D).

(3b) Definition, periodicity and continuity. For a kernel $h$ and $v\in\mathbb{R}^{d}$, the function $y\mapsto h(v+n(y))$ is Borel (a continuous function of the Borel map $y\mapsto v+n(y)$, Step 0(c)) and dominated by $M_{\lVert v\rVert}e$ by (D), hence integrable; put

$$H_{h}(v)=\int h\bigl(v+n(y)\bigr)\,dy\in\mathbb{R},\qquad \Theta=H_{h_{0}},\quad \Theta_{i}=H_{h_{i}},\quad \Theta_{ij}=H_{h_{ij}} .$$

Periodicity. For $k\in\mathbb{Z}^{d}$, $h(v+k+n(y))=h(v+n(y+k))$ by Step 0(c), and claim 3 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ with $a=k$, applied to the integrable function $y\mapsto h(v+n(y))$, gives $H_{h}(v+k)=\int h(v+n(y+k))\,dy=H_{h}(v)$. Continuity. Let $(v_{j})_{j\in\mathbb{N}}$ converge to $v$ in $(\mathbb{R}^{d},d_{E})$. Choose $J\in\mathbb{N}$ with $\lVert v_{j}-v\rVert<1$ for all $j\ge J$ (Convergent Sequence in a Metric Space, with $d_{E}(v_{j},v)=\lVert v_{j}-v\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), put $R=\lVert v\rVert+1$ and $w_{j}=v_{J+j}$, so that $\lVert w_{j}\rVert\le R$. For each $y$, $d_{E}(w_{j}+n(y),v+n(y))=\lVert w_{j}-v\rVert$, so $w_{j}+n(y)\to v+n(y)$ and $h(w_{j}+n(y))\to h(v+n(y))$ by claim 1 of Continuity Between Metric Spaces is Equivalent to Sequential Continuity. By (D) the functions $y\mapsto h(w_{j}+n(y))$ are all dominated by the integrable $M_{R}e$, so Dominated Convergence Theorem gives $H_{h}(w_{j})\to H_{h}(v)$; given $\varepsilon>0$ and $L$ with $|H_{h}(w_{j})-H_{h}(v)|<\varepsilon$ for $j\ge L$, we get $|H_{h}(v_{j})-H_{h}(v)|<\varepsilon$ for $j\ge J+L$. By claim 3 of Continuity Between Metric Spaces is Equivalent to Sequential Continuity, $H_{h}$ is continuous on $\mathbb{R}^{d}$; this is also continuity at every point in the sense of Continuity at a Point for Maps Between Euclidean Spaces, whose conditions $\sum_{i}(x_{i}-a_{i})^{2}<\delta^{2}$ and $(f(x)-f(a))^{2}<\varepsilon^{2}$ read $\lVert x-a\rVert<\delta$ and $|f(x)-f(a)|<\varepsilon$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Thus every $H_{h}$ belongs to $C_{\mathrm{per}}$.

(3c) Derivatives. Let $h$ be a kernel and $m\in[d]$ such that $\partial_{m}h$ is again a kernel (namely $h=h_{0}$, $\partial_{m}h=h_{m}$, or $h=h_{i}$, $\partial_{m}h=h_{im}$). We show that $\partial_{m}H_{h}(v)$ exists and equals $H_{\partial_{m}h}(v)$ for every $v$. Fix $v$; for real $t$ let $v[t]$ be the point with $m$th coordinate $t$ and the other coordinates those of $v$, as in Slice Function and the Partial Derivative, and let $U$ be the open interval with endpoints $v_{m}-1$ and $v_{m}+1$. For $t\in U$, $v[t]-v$ has the single nonzero coordinate $t-v_{m}$, so $\lVert v[t]\rVert\le\lVert v\rVert+1=:R$ (claims 1 and 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$). Apply Differentiation under the Integral Sign to $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\lambda_{d})$ and $f(t,y)=h(v[t]+n(y))$ on $U\times\mathbb{R}^{d}$. Condition (i) holds by (3b). For (ii), fix $y$ and $t_{0}\in U$ and put $b=v[t_{0}]+n(y)$; for real $\eta\ne0$ the point $v[t_{0}+\eta]+n(y)$ is $b$ with its $m$th coordinate $b_{m}$ replaced by $b_{m}+\eta$, so $(f(t_{0}+\eta,y)-f(t_{0},y))/\eta$ is exactly the difference quotient of Partial Derivative on a Euclidean Open Set for $h$ at $b$ in the $m$th variable; as $h$ is of class $C^{1}$, that definition's condition with value $\partial_{m}h(b)$ holds, and it is word for word the condition that $t\mapsto f(t,y)$ be differentiable at $t_{0}$ with derivative $\partial_{m}h(b)$ (Derivative at an Interior Point). For (iii), $|\partial_{m}h(v[t]+n(y))|\le M_{R}e(y)$ for $t\in U$ by (D), $\partial_{m}h$ being a kernel. The theorem shows that $F(t)=H_{h}(v[t])$ is differentiable on $U$ with $F'(t)=H_{\partial_{m}h}(v[t])$. Since the domain is all of $\mathbb{R}^{d}$, the radius $1$ is admissible in claim 1 of Slice Function and the Partial Derivative, $F$ is the slice function of $H_{h}$ at $v$ in the $m$th variable, and claim 2 there gives $\partial_{m}H_{h}(v)=F'(v_{m})=H_{\partial_{m}h}(v)$.

Consequently: $\Theta$ is continuous with partial derivatives $\partial_{i}\Theta=\Theta_{i}$ that exist everywhere and are continuous, so $\Theta$ is of class $C^{1}$ (clause 1 of C^k Maps on a Euclidean Open Set); likewise each $\Theta_{i}$ is of class $C^{1}$ with $\partial_{j}\Theta_{i}=\Theta_{ij}$; hence $\Theta$ is of class $C^{2}$ with $\partial_{j}\partial_{i}\Theta=\Theta_{ij}$ (clause 2 there, with $k=1$). By (3b) and Elementary Properties of Lattice-Periodic Functions §bounded each of the finitely many functions $\Theta$, $\Theta_{i}$, $\Theta_{ij}$ lies in $C_{\mathrm{per}}$ and is bounded; let $C_{\Theta}\ge0$ be a common bound (the largest of the individual bounds).

(3d) Positivity. Let $v\in\mathbb{R}^{d}$. The integrand $y\mapsto g_{s}(v+n(y))$ is positive everywhere. If $\Theta(v)=0$, then by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing it vanishes $\lambda_{d}$-almost everywhere, so $\mathbb{R}^{d}$ is $\lambda_{d}$-null, contradicting $\lambda_{d}(Q)=1$ and monotonicity. As $\Theta(v)\ge0$ by monotonicity of the integral, $\Theta(v)>0$.

(3e) Cell identities. For $w\in\mathbb{R}^{d}$ and Borel $B\subseteq Q$, and for $x\in\mathbb{R}^{d}$,

$$\int\mathbf{1}_{B}(x)\,\Theta(x-w)\,dx=\int\mathbf{1}_{B}\bigl(\pi(z)\bigr)\,g_{s}(z-w)\,dz,\qquad \int\mathbf{1}_{Q}(x)\,\Theta(x-w)\,dx=1,\qquad \int\mathbf{1}_{Q}(w)\,\Theta(x-w)\,dw=1 .$$

First identity. By Step 0(b) with $\lambda_{d}$ twice, the left side is $\int\bigl(\int\mathbf{1}_{B}(x)g_{s}(x+n(y)-w)\,dx\bigr)dy$. For fixed $y$, claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ with $a=n(y)$, applied to the nonnegative Borel function $f_{y}(z)=\mathbf{1}_{B}(z-n(y))g_{s}(z-w)$, for which $f_{y}(x+n(y))=\mathbf{1}_{B}(x)g_{s}(x+n(y)-w)$, shows that the inner integral is $\int\mathbf{1}_{B}(z-n(y))g_{s}(z-w)\,dz$. Exchanging the order again by Step 0(b), taking the factor $g_{s}(z-w)$ out of the inner integral (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) and using Step 0(c2), the left side equals $\int\bigl(\int\mathbf{1}_{B}(z-n(y))\,dy\bigr)g_{s}(z-w)\,dz=\int\mathbf{1}_{B}(\pi(z))g_{s}(z-w)\,dz$. Second identity. Take $B=Q$: $\pi(z)\in Q$ for every $z$ (The Half-Open Unit Cell Tiles Euclidean Space §wrap), so the right side is $\int g_{s}(z-w)\,dz=1$ (Conventions). Third identity. By Step 0(b) the left side is $\int\bigl(\int\mathbf{1}_{Q}(w)g_{s}(x+n(y)-w)\,dw\bigr)dy$. For fixed $y$ put $a=x+n(y)$; claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$, applied to $f(u)=\mathbf{1}_{Q}(a-u)g_{s}(u)$, for which $f(a-w)=\mathbf{1}_{Q}(w)g_{s}(a-w)$, shows that the inner integral is $\int\mathbf{1}_{Q}(x-u+n(y))g_{s}(u)\,du$. Exchanging the order by Step 0(b) and using Step 0(c3) with $z=x-u$, the left side is $\int\bigl(\int\mathbf{1}_{Q}(x-u+n(y))\,dy\bigr)g_{s}(u)\,du=\int g_{s}(u)\,du=1$.

Step 4 (Clause 3). Define $p:\mathbb{R}^{d}\to\mathbb{R}$ by

$$p(x)=\int\Theta(x-w)\,\mu(dw),$$

which is $\Theta*\mu$ in the notation of The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §continuous, $\Theta$ being continuous and bounded by $C_{\Theta}$.

(4a) Regularity. By The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §derivative with $G=\Theta$ and $C=C_{\Theta}$ (legitimate by (3c)), $p$ is of class $C^{1}$ with $\partial_{i}p=\Theta_{i}*\mu$. By the same clause with $G=\Theta_{i}$, which is of class $C^{1}$ with $|\Theta_{i}|\le C_{\Theta}$ and $|\partial_{j}\Theta_{i}|=|\Theta_{ij}|\le C_{\Theta}$, the function $\partial_{i}p$ is of class $C^{1}$ with $\partial_{j}\partial_{i}p=\Theta_{ij}*\mu$. By clause 2 of C^k Maps on a Euclidean Open Set with $k=1$, $p$ is of class $C^{2}$ on $\mathbb{R}^{d}$.

(4b) Periodicity, positivity and a bound. For $k\in\mathbb{Z}^{d}$, $p(x+k)=\int\Theta(x-w+k)\,\mu(dw)=p(x)$ by the periodicity of $\Theta$ (3b). Also $0\le p(x)\le C_{\Theta}$ by monotonicity of the integral. If $p(x)=0$, then The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing, applied in $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ to the nonnegative Borel function $w\mapsto\Theta(x-w)$, which is positive everywhere by (3d), would make $\mathbb{R}^{d}$ a $\mu$-null set, contradicting $\mu(\mathbb{R}^{d})=1$. Hence $p(x)>0$ for every $x$, and $p\in C^{2}_{\mathrm{per}}$.

(4c) The density. Let $B\in\mathcal{B}(\mathbb{R}^{d})$ with $B\subseteq Q$. Taking the constant $\mathbf{1}_{B}(x)$ inside the integral defining $p(x)$ (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), using Step 0(b) with the measures $\lambda_{d}$ and $\mu$, and then (3e),

$$\int\mathbf{1}_{B}\,p\,d\lambda_{d}=\int\Bigl(\int\mathbf{1}_{B}(x)\Theta(x-w)\,dx\Bigr)\mu(dw)=\int\Bigl(\int\mathbf{1}_{B}\bigl(\pi(z)\bigr)g_{s}(z-w)\,dz\Bigr)\mu(dw).$$

With $\varphi_{B}=\mathbf{1}_{B}\circ\pi$ as in (2b), the evenness of $g_{s}$ gives $\int\mathbf{1}_{B}(\pi(z))g_{s}(z-w)\,dz=\int g_{s}(w-z)\varphi_{B}(z)\,dz=(g_{s}*\varphi_{B})(w)$, so by (W) of (2b), with $\kappa=\mu$,

$$\int\mathbf{1}_{B}\,p\,d\lambda_{d}=\int g_{s}*\varphi_{B}\,d\mu=\pi_{\#}\mu_{s}(B)=S_{s}\mu(B).$$

Now let $A\in\mathcal{B}(\mathbb{R}^{d})$ be arbitrary. As $S_{s}\mu\in\mathcal{P}(\mathbb{T}^{d})$ (The Heat Semigroup on the Probability Measures on the Torus §heat), Step 0(a) and the case $B=A\cap Q$ give

$$S_{s}\mu(A)=S_{s}\mu(A\cap Q)=\int\mathbf{1}_{A\cap Q}\,p\,d\lambda_{d}=\int\mathbf{1}_{A}\,(\mathbf{1}_{Q}p)\,d\lambda_{d}.$$

The function $\mathbf{1}_{Q}p$ is Borel (a product of Borel functions, $p$ being continuous) and nonnegative, so it is a density of $S_{s}\mu$ with respect to $\lambda_{d}$ (The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities). Together with (4a) and (4b) this proves clause 3.

Step 5 (Clause 4).

(5a) The logarithm of the density. The set $V=(0,\infty)$ is open in $\mathbb{R}^{1}$, $p$ takes values in $V$ by (4b), and $\log$ is smooth on $V$ with $\log'(t)=t^{-1}$ (The Natural Logarithm), hence of class $C^{2}$ (Smooth Map on a Euclidean Open Set), with $\partial_{1}\log(t)=t^{-1}$ by claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line. By claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, $\ell=\log\circ p$ is of class $C^{2}$ on $\mathbb{R}^{d}$; it is $\mathbb{Z}^{d}$-periodic because $p$ is, so $\ell\in C^{2}_{\mathrm{per}}$; and by claim 1 there

$$\partial_{i}\ell(x)=p(x)^{-1}\,\partial_{i}p(x)\qquad(x\in\mathbb{R}^{d},\ i\in[d]). \qquad(\mathrm{L})$$

By claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and Elementary Properties of Lattice-Periodic Functions §algebra, $p\,\ell\in C^{2}_{\mathrm{per}}$; it is continuous (claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), hence bounded by some $M_{\phi}\ge0$ (Elementary Properties of Lattice-Periodic Functions §bounded), and $p(x)\ell(x)=\phi(p(x))$ for every $x$.

(5b) Finite entropy. By clause 3, $\mathbf{1}_{Q}p$ is a density of $S_{s}\mu$. Since $\phi(0)=0$, $\phi\circ(\mathbf{1}_{Q}p)=\mathbf{1}_{Q}\,(p\,\ell)$, a Borel function with $|\phi\circ(\mathbf{1}_{Q}p)|\le M_{\phi}\mathbf{1}_{Q}$, whose integral is $M_{\phi}\lambda_{d}(Q)=M_{\phi}<\infty$ (claim 1 of Linearity and Monotonicity of the Lebesgue Integral). So $\phi\circ(\mathbf{1}_{Q}p)$ is integrable, $S_{s}\mu$ has finite entropy by The Entropy of a Probability Measure on Euclidean Space §entropy, and $\mathrm{Ent}(S_{s}\mu)=\int\mathbf{1}_{Q}\,\phi\circ p\,d\lambda_{d}$.

(5c) A supporting line. For real $t>0$ and $r\ge0$,

$$\phi(r)\ge\phi(t)+(1+\log t)(r-t). \qquad(\mathrm{S})$$

If $r=0$ the difference of the two sides is $0-t\log t+(1+\log t)t=t\ge0$. If $r>0$, then $\log r=\log(r/t)+\log t$ (The Natural Logarithm), and the difference is $r\log r-t\log t-(1+\log t)(r-t)=r\log(r/t)-r+t\ge r(1-t/r)-r+t=0$, by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log applied to $r/t$ and $r>0$.

(5d) Monotonicity. Suppose $\mu$ has finite entropy and choose, by The Entropy of a Probability Measure on Euclidean Space §entropy, a density $\rho$ of $\mu$ with $\phi\circ\rho$ integrable. Put $\rho_{Q}=\mathbf{1}_{Q}\rho$. For Borel $A$, Step 0(a) gives $\mu(A)=\mu(A\cap Q)=\int\mathbf{1}_{A}\rho_{Q}\,d\lambda_{d}$, so $\rho_{Q}$ is a density of $\mu$; and $\phi\circ\rho_{Q}=\mathbf{1}_{Q}(\phi\circ\rho)$ (as $\phi(0)=0$), Borel by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous with $|\phi\circ\rho_{Q}|\le|\phi\circ\rho|$, so it is integrable and $\mathrm{Ent}(\mu)=\int\phi\circ\rho_{Q}\,d\lambda_{d}$, the entropy not depending on the density (The Entropy of a Probability Measure on Euclidean Space §entropy).

Fix $x\in\mathbb{R}^{d}$ and put $K_{x}(w)=\mathbf{1}_{Q}(w)\Theta(x-w)$, a Borel function with $0\le K_{x}\le C_{\Theta}\mathbf{1}_{Q}$ and $\int K_{x}\,dw=1$ by (3e). Since $w\mapsto\Theta(x-w)$ is bounded and Borel, claim 3 of Image Measures, Measures with Densities, and Change of Variables (integrable case) and $\mathbf{1}_{Q}\rho_{Q}=\rho_{Q}$ give

$$p(x)=\int\Theta(x-w)\rho_{Q}(w)\,dw=\int K_{x}\,\rho_{Q}\,dw .$$

Apply (S) with $t=p(x)>0$ and $r=\rho_{Q}(w)$ and multiply by $K_{x}(w)\ge0$:

$$K_{x}(w)\,\phi(\rho_{Q}(w))\ge\phi(p(x))\,K_{x}(w)+\bigl(1+\log p(x)\bigr)\bigl(K_{x}(w)\rho_{Q}(w)-p(x)K_{x}(w)\bigr)\qquad(w\in\mathbb{R}^{d}).$$

All functions of $w$ here are integrable: $K_{x}\le C_{\Theta}\mathbf{1}_{Q}$, $K_{x}\rho_{Q}\le C_{\Theta}\rho_{Q}$ and $|K_{x}\,\phi\circ\rho_{Q}|\le C_{\Theta}|\phi\circ\rho_{Q}|$. Integrating in $w$ (claim 2 of Linearity and Monotonicity of the Lebesgue Integral) and using $\int K_{x}=1$ and $\int K_{x}\rho_{Q}=p(x)$,

$$\int K_{x}\,\phi\circ\rho_{Q}\,dw\ge\phi(p(x)). \qquad(\mathrm{J})$$

Let $\Psi(x,w)=\mathbf{1}_{Q}(x)\mathbf{1}_{Q}(w)\Theta(x-w)\,\phi(\rho_{Q}(w))$ and let $\Psi^{+}$, $\Psi^{-}$ be its positive and negative parts, $\Psi^{\pm}(x,w)=\mathbf{1}_{Q}(x)\mathbf{1}_{Q}(w)\Theta(x-w)\,(\phi\circ\rho_{Q})^{\pm}(w)$, both of the form of Step 0(b). By that step, taking $(\phi\circ\rho_{Q})^{\pm}(w)$ out of the inner integral and using (3e),

$$\int\Bigl(\int\Psi^{\pm}(x,w)\,dw\Bigr)dx=\int(\phi\circ\rho_{Q})^{\pm}(w)\,\mathbf{1}_{Q}(w)\Bigl(\int\mathbf{1}_{Q}(x)\Theta(x-w)\,dx\Bigr)dw=\int(\phi\circ\rho_{Q})^{\pm}\,d\lambda_{d}<\infty,$$

where $\mathbf{1}_{Q}(\phi\circ\rho_{Q})^{\pm}=(\phi\circ\rho_{Q})^{\pm}$ because $\phi\circ\rho_{Q}$ vanishes off $Q$. The functions $G^{\pm}(x)=\int\Psi^{\pm}(x,w)\,dw$ are measurable by Step 0(b), finite at every $x$ (each is at most $C_{\Theta}\int|\phi\circ\rho_{Q}|\,d\lambda_{d}$), and have finite integrals by the display; hence they are integrable, and $G^{+}(x)-G^{-}(x)=\mathbf{1}_{Q}(x)\int K_{x}\,\phi\circ\rho_{Q}\,dw$. By (J), $\mathbf{1}_{Q}(x)\,\phi(p(x))\le G^{+}(x)-G^{-}(x)$ for every $x$ (both sides are $0$ off $Q$). Monotonicity and linearity of the integral (claim 2 of Linearity and Monotonicity of the Lebesgue Integral), (5b) and the display give

$$\mathrm{Ent}(S_{s}\mu)=\int\mathbf{1}_{Q}\,\phi\circ p\,d\lambda_{d}\le\int G^{+}\,d\lambda_{d}-\int G^{-}\,d\lambda_{d}=\int(\phi\circ\rho_{Q})^{+}\,d\lambda_{d}-\int(\phi\circ\rho_{Q})^{-}\,d\lambda_{d}=\mathrm{Ent}(\mu).$$

Step 6 (Clause 5). Let $\ell=\log\circ p\in C^{2}_{\mathrm{per}}$ be as in (5a), and let $r:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the map $x\mapsto p(x)^{-1}\nabla p(x)$; by (L), $r=\nabla\ell$. Since $\ell$ is of class $C^{2}$, it is of class $C^{1}$ (claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), so $\ell\in C^{1}_{\mathrm{per}}$ and, by Optimal Transport on the Flat Torus: Standing Notation §calculus, $r$ is Borel with bounded components in $C_{\mathrm{per}}$ and its class belongs to $L^{2}(S_{s}\mu;\mathbb{R}^{d})$. Write $\sigma=S_{s}\mu$.

(6a) The integration-by-parts identity. Let $f\in C^{\infty}_{\mathrm{per}}$. For $i\in[d]$, $\partial_{i}f\in C^{\infty}_{\mathrm{per}}$ and $\partial_{i}\partial_{i}f\in C_{\mathrm{per}}$ (Elementary Properties of Lattice-Periodic Functions §derivative), and $p\in C^{1}_{\mathrm{per}}$ (claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous); the products $(\partial_{i}\partial_{i}f)\,p$ and $(\partial_{i}f)(\partial_{i}\ell)\,p$ lie in $C_{\mathrm{per}}$ (Elementary Properties of Lattice-Periodic Functions §algebra). For every $u\in C_{\mathrm{per}}$, Continuous Periodic Functions are Power-Integrable and Dense on the Torus §integral gives that $\mathbf{1}_{Q}u$ is integrable with $\int_{\mathbb{T}^{d}}u|_{Q}\,dx=\int\mathbf{1}_{Q}u\,d\lambda_{d}$, and, if $u=vp$ with $v$ bounded Borel, claim 3 of Image Measures, Measures with Densities, and Change of Variables (the density being $\mathbf{1}_{Q}p$ by clause 3) gives $\int\mathbf{1}_{Q}u\,d\lambda_{d}=\int v\,d\sigma$. Now $\Delta f=\sum_{i=1}^{d}\partial_{i}\partial_{i}f$ (The Laplacian of a Twice Continuously Differentiable Function §laplacian) is bounded and Borel (Optimal Transport on the Flat Torus: Standing Notation §calculus). Using linearity of the integral, Vanishing Mean of a Derivative and Integration by Parts on the Torus §parts with $g=\partial_{i}f$ and $h=p$, and $\partial_{i}p=p\,\partial_{i}\ell$ from (L),

$$\int\Delta f\,d\sigma=\sum_{i=1}^{d}\int_{\mathbb{T}^{d}}(\partial_{i}\partial_{i}f)\,p\,dx=-\sum_{i=1}^{d}\int_{\mathbb{T}^{d}}(\partial_{i}f)(\partial_{i}p)\,dx=-\sum_{i=1}^{d}\int_{\mathbb{T}^{d}}(\partial_{i}f)(\partial_{i}\ell)\,p\,dx=-\int\nabla f\cdot r\,d\sigma=-\langle r,\nabla f\rangle_{\sigma}.$$

(6b) Finite torus Fisher information. By (6a) and the Cauchy--Schwarz inequality in the real Hilbert space $L^{2}(\sigma;\mathbb{R}^{d})$ (Optimal Transport on the Flat Torus: Standing Notation §fields, The Cauchy-Schwarz Inequality in a Real Inner Product Space), $|\int\Delta f\,d\sigma|\le\lVert r\rVert_{\sigma}\lVert\nabla f\rVert_{\sigma}$ for every $f\in C^{\infty}_{\mathrm{per}}$. With the real number $C=\lVert r\rVert_{\sigma}\ge0$, Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §finite gives $\sigma\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$.

(6c) The class of $r$ is tangent. By Existence of Mollifier Kernels of Every Radius choose a mollifier kernel $\varrho$ of radius $1$ on $\mathbb{R}^{d}$, and for $\varepsilon>0$ let $\varrho_{\varepsilon}$ be its rescaling (Rescaling a Mollifier Kernel), a mollifier kernel of radius $\varepsilon$. By Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$ it is smooth and vanishes outside the ball of radius $\varepsilon$, and $\ell|_{Q}\in\mathcal{L}^{1}(\mathbb{T}^{d})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member; so $f_{\varepsilon}=\varrho_{\varepsilon}\star(\ell|_{Q})$ belongs to $C^{\infty}_{\mathrm{per}}$ by Properties of Periodic Convolution on the Torus §smooth. For $k\in\mathbb{N}$ let $t_{k}$ be the image of $k$ in $\mathbb{R}$ (The Canonical Map from the Natural Numbers to a Field) and $\eta_{k}=t_{k}^{-1}d^{-1}>0$. By Transfer of a Partial Derivative in Periodic Convolution §uniform-c1, applied to $\varphi=\ell\in C^{1}_{\mathrm{per}}$ and $\eta=\eta_{k}$, there is $\varepsilon_{0}>0$ with $|\partial_{i}f_{\varepsilon}(x)-\partial_{i}\ell(x)|\le\eta_{k}$ for all $x$, all $i$ and all $0<\varepsilon<\varepsilon_{0}$; choose such an $\varepsilon$, say $\varepsilon=\varepsilon_{0}/2$, and put $F_{k}=f_{\varepsilon}$. Then $\lVert\nabla F_{k}(x)-r(x)\rVert^{2}\le d\,\eta_{k}^{2}=t_{k}^{-2}d^{-1}\le t_{k}^{-2}$ for every $x$, so, $\sigma$ being a probability measure, $\lVert\nabla F_{k}-r\rVert_{\sigma}\le t_{k}^{-1}$ (monotonicity of the integral and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). As $(t_{k}^{-1})_{k\in\mathbb{N}}$ converges to $0$ (claim 3 of The Archimedean Property of the Real Numbers), the classes of $\nabla F_{k}$, which belong to $G_{\sigma}$ (The Tangent Space of the Torus Wasserstein Space at a Probability Measure §gradients), converge to the class of $r$ in $L^{2}(\sigma;\mathbb{R}^{d})$ with the distance of its norm. By Sequential Characterization of the Closure in a Metric Space, the class of $r$ lies in the closure of $G_{\sigma}$, which is $T_{\sigma}$ (The Tangent Space of the Torus Wasserstein Space at a Probability Measure §tangent).

(6d) Identification of the score. By (6c) the class of $r$ belongs to $T_{\sigma}$ and by (6a) it satisfies $\langle r,\nabla f\rangle_{\sigma}=-\int\Delta f\,d\sigma$ for every $f\in C^{\infty}_{\mathrm{per}}$. By Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §score the score $\xi_{\sigma}$ is the unique member of $T_{\sigma}$ with this property, so $\xi_{S_{s}\mu}$ is the class of $x\mapsto p(x)^{-1}\nabla p(x)$. $\blacksquare$