Proof of Heat Regularisation of an Intrinsic Test Function: a Laplacian Test Function whose Laplacian is Minus the Pairing of the Gradient with the Score

Each result cited below is universally quantified over the data in its own statement.

For clauses 1 and 2 fix $s$ with $0<s\le\tfrac12$, write $\Theta=\Theta_{s}$ and $p_{\mu}=p^{s}_{\mu}$ for $\mu\in\mathcal{P}(\mathbb{T}^{d})$, and let $c_{s},C_{s}$ be the constants of The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §regularity and $L_{s}$ that of The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §lipschitz. As in Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings §average, $\int_{Q}f\,d\lambda_{d}$ stands for $\int\mathbf{1}_{Q}f\,d\lambda_{d}$; since $\lambda_{d}(Q)=1$ by The Half-Open Unit Cell Tiles Euclidean Space §cell, $|\int_{Q}f\,d\lambda_{d}|\le M$ whenever $f$ is Borel with $|f|\le M$ on $Q$.

Step 0 (Kernel estimates). Let $\Psi$ be one of the $d+1$ functions $\Theta,\partial_{1}\Theta,\dots,\partial_{d}\Theta$. By The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §regularity, $\Theta\in C^{2}_{\mathrm{per}}$, so $\Psi$ is $\mathbb{Z}^{d}$-periodic and of class $C^{1}$ on $\mathbb{R}^{d}$ by Elementary Properties of Lattice-Periodic Functions §derivative; by the same clause of the kernel lemma, $|\Psi|\le C_{s}$ and every first partial derivative of $\Psi$ (namely $\partial_{k}\Theta$, respectively $\partial_{k}\partial_{j}\Theta$) is bounded by $C_{s}$ in absolute value. The Lipschitz bound (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder, applied on the open set $\mathbb{R}^{d}$, therefore gives

$$|\Psi(u)-\Psi(u')|\le\sqrt{d}\,C_{s}\,\lVert u-u'\rVert\qquad(u,u'\in\mathbb{R}^{d}).\tag{0.1}$$

For $\mu\in\mathcal{P}(\mathbb{T}^{d})$, The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §density gives $p_{\mu}\in C^{2}_{\mathrm{per}}$, $c_{s}\le p_{\mu}\le C_{s}$, and $|\partial_{k}p_{\mu}(y)|=|\int\partial_{k}\Theta(y-x)\,\mu(dx)|\le C_{s}$; the same Taylor bound gives

$$|p_{\mu}(y)-p_{\mu}(y')|\le\sqrt{d}\,C_{s}\,\lVert y-y'\rVert\qquad(y,y'\in\mathbb{R}^{d}).\tag{0.2}$$

Periodic reduction. If $f:\mathbb{R}^{d}\to\mathbb{R}$ is $\mathbb{Z}^{d}$-periodic and $|f(u)-f(u')|\le M\lVert u-u'\rVert$ for all $u,u'$, then $|f(u)-f(u')|\le M\,d_{\mathbb{T}}(u,u')$ for all $u,u'$. Indeed $k=u'-u-\varpi(u'-u)\in\mathbb{Z}^{d}$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, so $f(u')=f(u'-k)$, and $\lVert u-(u'-k)\rVert=\lVert\varpi(u'-u)\rVert=d_{\mathbb{T}}(u,u')$ by The Wrapped Displacement and the Flat Torus Distance §distance.

Weights. For $\mu\in\mathcal{P}(\mathbb{T}^{d})$, $x\in\mathbb{R}^{d}$ and $\Psi$ as above put $w^{\Psi}_{\mu,x}(y)=\Psi(y-x)/p_{\mu}(y)$. This function is continuous, $\mathbb{Z}^{d}$-periodic (as $\Psi$ and $p_{\mu}$ are), hence Borel, and $|w^{\Psi}_{\mu,x}|\le C_{s}/c_{s}$. Writing

$$w^{\Psi}_{\mu,x}(y)-w^{\Psi}_{\mu,x}(y')=\frac{\Psi(y-x)-\Psi(y'-x)}{p_{\mu}(y)}+\Psi(y'-x)\,\frac{p_{\mu}(y')-p_{\mu}(y)}{p_{\mu}(y)\,p_{\mu}(y')}$$

and using (0.1), (0.2) and then the periodic reduction, we get, with $M_{s}=\sqrt{d}\,C_{s}/c_{s}+\sqrt{d}\,C_{s}^{2}/c_{s}^{2}$,

$$|w^{\Psi}_{\mu,x}(y)-w^{\Psi}_{\mu,x}(y')|\le M_{s}\,d_{\mathbb{T}}(y,y')\qquad(y,y'\in\mathbb{R}^{d}).\tag{0.3}$$

Similarly, for $\mu,\mu'\in\mathcal{P}(\mathbb{T}^{d})$ and $x,x'\in\mathbb{R}^{d}$, splitting $w^{\Psi}_{\mu',x'}(y)-w^{\Psi}_{\mu,x}(y)$ as $\bigl(\Psi(y-x')-\Psi(y-x)\bigr)/p_{\mu'}(y)+\Psi(y-x)\bigl(p_{\mu}(y)-p_{\mu'}(y)\bigr)/\bigl(p_{\mu'}(y)p_{\mu}(y)\bigr)$ and using (0.1) and the first inequality of The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §lipschitz,

$$\sup_{y\in\mathbb{R}^{d}}|w^{\Psi}_{\mu',x'}(y)-w^{\Psi}_{\mu,x}(y)|\le\frac{\sqrt{d}\,C_{s}}{c_{s}}\lVert x'-x\rVert+\frac{C_{s}L_{s}}{c_{s}^{2}}\,W_{\mathbb{T}}(\mu',\mu).\tag{0.4}$$

Step 1 (The candidate field; condition (a) and tangency). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$. By Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §measures, $S_{s}\mu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, so by condition (b) of intrinsic test functions on $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ the function $\varphi$ is differentiable along couplings at $S_{s}\mu$ and $\nabla\varphi(S_{s}\mu)\in T_{S_{s}\mu}$. Choose a Borel representative $F_{\mu}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ of $\nabla\varphi(S_{s}\mu)$; then $\int\lVert F_{\mu}\rVert^{2}\,d(S_{s}\mu)<\infty$. Define $g(\mu,x)=A^{\mu}_{s}F_{\mu}(x)=\int_{Q}F_{\mu}(y)\,\Theta(y-x)\,\lambda_{d}(dy)$. By Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings §average this does not depend on the representative, so $g:\mathcal{P}(\mathbb{T}^{d})\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ is well defined and is the map $(\mu,x)\mapsto A^{\mu}_{s}\nabla\varphi(S_{s}\mu)(x)$ of the statement. By the same clause each $g_{i}(\mu,\cdot)$ belongs to $C^{1}_{\mathrm{per}}$, which is condition (a) of a gradient field, and for all $x\in\mathbb{R}^{d}$ and $i,j\in[d]$

$$g_{i}(\mu,x)=\int_{Q}F_{\mu,i}(y)\,\Theta(y-x)\,\lambda_{d}(dy),\qquad\partial_{j}g_{i}(\mu,x)=-\int_{Q}F_{\mu,i}(y)\,\partial_{j}\Theta(y-x)\,\lambda_{d}(dy).\tag{1.1}$$

Since the class of $F_{\mu}$ lies in $T_{S_{s}\mu}$, Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings §tangent shows that the class of $g(\mu,\cdot)$ in $L^{2}(\mu;\mathbb{R}^{d})$ lies in $T_{\mu}$.

Step 2 (Condition (b): differentiability along couplings). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $\varepsilon>0$. By Step 1 and Differentiability Along Couplings of a Function on the Torus Wasserstein Space, and Its Gradient §gradient, $\nabla\varphi(S_{s}\mu)$ has the property of Differentiability Along Couplings of a Function on the Torus Wasserstein Space, and Its Gradient §differentiable at $S_{s}\mu$; take the corresponding $\theta>0$ for this $\varepsilon$. Let $\nu\in\mathcal{P}(\mathbb{T}^{d})$ and $\gamma\in\Pi(\mu,\nu)$ with $I_{\mathbb{T}}(\gamma)<\theta^{2}$, and let $\gamma^{s}$ be the synchronous coupling of Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings §synchronous, formed with $F=F_{\mu}$. That clause gives $\gamma^{s}\in\Pi(S_{s}\mu,S_{s}\nu)$, $I_{\mathbb{T}}(\gamma^{s})=I_{\mathbb{T}}(\gamma)<\theta^{2}$ and $\mathcal{J}_{\mathbb{T}}(F_{\mu},\gamma^{s})=\mathcal{J}_{\mathbb{T}}(g(\mu,\cdot),\gamma)$. Hence

$$\bigl|\varphi(S_{s}\nu)-\varphi(S_{s}\mu)-\mathcal{J}_{\mathbb{T}}(g(\mu,\cdot),\gamma)\bigr|=\bigl|\varphi(S_{s}\nu)-\varphi(S_{s}\mu)-\mathcal{J}_{\mathbb{T}}(F_{\mu},\gamma^{s})\bigr|\le\varepsilon\sqrt{I_{\mathbb{T}}(\gamma^{s})}=\varepsilon\sqrt{I_{\mathbb{T}}(\gamma)}.$$

So $\varphi\circ S_{s}$ is differentiable along couplings at $\mu$ with gradient the class of $g(\mu,\cdot)$, and by the uniqueness in Differentiability Along Couplings of a Function on the Torus Wasserstein Space, and Its Gradient §gradient, $\nabla(\varphi\circ S_{s})(\mu)$ is the class of $g(\mu,\cdot)$, that is $A^{\mu}_{s}\nabla\varphi(S_{s}\mu)$. This is condition (b) of a gradient field.

Step 3 (Condition (c): joint continuity). Let $(\mu_{n})$ converge to $\mu$ in $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$ and $(x_{n})$ converge to $x$ in $\mathbb{R}^{d}$, and fix $i\in[d]$ and $\Psi\in\{\Theta,\partial_{1}\Theta,\dots,\partial_{d}\Theta\}$. For $\mu'\in\mathcal{P}(\mathbb{T}^{d})$ and $x'\in\mathbb{R}^{d}$ put $h(\mu',x')=\int_{Q}F_{\mu',i}(y)\,\Psi(y-x')\,\lambda_{d}(dy)$. By (1.1), $g_{i}=h$ when $\Psi=\Theta$ and $\partial_{j}g_{i}=-h$ when $\Psi=\partial_{j}\Theta$, so it suffices to show that $h(\mu_{n},x_{n})\to h(\mu,x)$.

Write $\nu_{n}=S_{s}\mu_{n}$, $\nu=S_{s}\mu$, $F_{n}=F_{\mu_{n}}$, $F=F_{\mu}$, $w_{n}=w^{\Psi}_{\mu_{n},x_{n}}$ and $w=w^{\Psi}_{\mu,x}$. By Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information §contraction, $W_{\mathbb{T}}(\nu_{n},\nu)\le W_{\mathbb{T}}(\mu_{n},\mu)\to0$. For each $n$ choose, by The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §optimal, an optimal coupling $\gamma_{n}\in\Pi(\nu_{n},\nu)$, so that $I_{\mathbb{T}}(\gamma_{n})=W_{\mathbb{T}}(\nu_{n},\nu)^{2}\to0$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §optimal. All $\nu_{n}$ and $\nu$ lie in $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ by Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §measures, so condition (c) of intrinsic test functions on $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ gives

$$D_{n}=\int\lVert F_{n}(\mathrm{pr}_{1}(w'))-F(\mathrm{pr}_{2}(w'))\rVert^{2}\,\gamma_{n}(dw')\longrightarrow0 .$$

Since $|w_{n}|\le C_{s}/c_{s}$ and $\int|F_{n,i}|\,d\nu_{n}\le\lVert F_{n}\rVert_{\nu_{n}}<\infty$ (Cauchy–Schwarz, Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm), the function $F_{n,i}w_{n}$ is integrable with respect to $\nu_{n}$. As $\mathbf{1}_{Q}p_{\mu_{n}}$ is a density of $\nu_{n}$ by The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §density, claim 3 of Image Measures, Measures with Densities, and Change of Variables gives $\int F_{n,i}w_{n}\,d\nu_{n}=\int\mathbf{1}_{Q}F_{n,i}\,\Psi(\cdot-x_{n})\,d\lambda_{d}=h(\mu_{n},x_{n})$, and the change-of-variables formula with $(\mathrm{pr}_{1})_{\#}\gamma_{n}=\nu_{n}$ gives $h(\mu_{n},x_{n})=\int F_{n,i}(\mathrm{pr}_{1}(w'))\,w_{n}(\mathrm{pr}_{1}(w'))\,\gamma_{n}(dw')$. In the same way, with $(\mathrm{pr}_{2})_{\#}\gamma_{n}=\nu$, $h(\mu,x)=\int F_{i}(\mathrm{pr}_{2}(w'))\,w(\mathrm{pr}_{2}(w'))\,\gamma_{n}(dw')$. Abbreviating $y=\mathrm{pr}_{1}(w')$, $y'=\mathrm{pr}_{2}(w')$ and splitting $F_{n,i}(y)w_{n}(y)-F_{i}(y')w(y')$ as

$$\bigl(F_{n,i}(y)-F_{i}(y')\bigr)w_{n}(y)+F_{i}(y')\bigl(w_{n}(y)-w(y)\bigr)+F_{i}(y')\bigl(w(y)-w(y')\bigr),$$

we bound the three integrals against $\gamma_{n}$. The first is at most $(C_{s}/c_{s})\int\lVert F_{n}(y)-F(y')\rVert\,d\gamma_{n}\le(C_{s}/c_{s})\sqrt{D_{n}}$ by Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm. The second is at most $\delta_{n}\int|F_{i}(y')|\,d\gamma_{n}=\delta_{n}\int|F_{i}|\,d\nu\le\delta_{n}\lVert F\rVert_{\nu}$ by Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm, where $\delta_{n}$ is the right side of (0.4) with $(\mu',x')=(\mu_{n},x_{n})$. The third is at most $M_{s}\int|F_{i}(y')|\,d_{\mathbb{T}}(y,y')\,d\gamma_{n}\le M_{s}\lVert F\rVert_{\nu}\sqrt{I_{\mathbb{T}}(\gamma_{n})}$ by (0.3), Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm and Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost. Hence

$$|h(\mu_{n},x_{n})-h(\mu,x)|\le\frac{C_{s}}{c_{s}}\sqrt{D_{n}}+\delta_{n}\lVert F\rVert_{\nu}+M_{s}\lVert F\rVert_{\nu}\sqrt{I_{\mathbb{T}}(\gamma_{n})},$$

and each term tends to $0$, because $D_{n}\to0$, $I_{\mathbb{T}}(\gamma_{n})\to0$, $\lVert x_{n}-x\rVert\to0$ and $W_{\mathbb{T}}(\mu_{n},\mu)\to0$. This is condition (c) of a gradient field.

Step 4 (Clause 1). By Steps 1–3, $g$ is a gradient field of $\varphi\circ S_{s}$, and the class of $g(\mu,\cdot)$ lies in $T_{\mu}$ for every $\mu$ by Step 1. By Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes §criterion, $\varphi\circ S_{s}$ is a Laplacian test function; by Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes §unique its gradient field is $g$; and Step 2 gives $\nabla(\varphi\circ S_{s})(\mu)=A^{\mu}_{s}\nabla\varphi(S_{s}\mu)$.

Step 5 (Clause 2). Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$. By The Laplacian of a Laplacian Test Function on the Torus Wasserstein Space §laplacian and Step 4, $\mathcal{L}(\varphi\circ S_{s})(\mu)=\int\sum_{i=1}^{d}\partial_{i}g_{i}(\mu,x)\,\mu(dx)$. Since $g(\mu,\cdot)$ is the heat average $a$ of $F_{\mu}$, Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings §divergence gives $\int\sum_{i}\partial_{i}g_{i}(\mu,\cdot)\,d\mu=-\langle F_{\mu},\xi_{S_{s}\mu}\rangle_{S_{s}\mu}=-\langle\nabla\varphi(S_{s}\mu),\xi_{S_{s}\mu}\rangle_{S_{s}\mu}$.

Step 6 (Clause 3). Let $\mu$, $(\mu_{n})$ and $(s_{n})$ be as in clause 3, and write $N_{n}=N_{s_{n}}$ for the Gaussian measure of Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings at time $s_{n}$. For each $n$, $S_{s_{n}}\mu_{n}\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ by Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §measures; let $F_{n}$ be a Borel representative of $\nabla\varphi(S_{s_{n}}\mu_{n})$ and $G$ one of $\nabla\varphi(\mu)$, and let $a_{n}(x)=\int_{Q}F_{n}(y)\,\Theta_{s_{n}}(y-x)\,\lambda_{d}(dy)$, which represents $A^{\mu_{n}}_{s_{n}}\nabla\varphi(S_{s_{n}}\mu_{n})$ by Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §heat-average. In this order: choose, by The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist §optimal, an optimal coupling $\beta_{n}\in\Pi(\mu_{n},\mu)$, so $I_{\mathbb{T}}(\beta_{n})=W_{\mathbb{T}}(\mu_{n},\mu)^{2}$; put $P_{n}=\beta_{n}\boxtimes N_{n}\in\mathcal{P}(\mathbb{R}^{d+d+d})$, the product measure read with $q=d+d$ and $p=d$; and let $\kappa_{n}=\sigma_{\#}P_{n}$, where $\sigma:\mathbb{R}^{d+d+d}\to\mathbb{R}^{d+d}$ is given by $\sigma(\iota^{d+d,d}(w',z))=\iota^{d,d}\bigl(\pi(\mathrm{pr}_{1}(w')+z),\mathrm{pr}_{2}(w')\bigr)$. The map $\sigma$ is Borel, being the pairing of the first component map $\pi\circ(\mathrm{pr}_{1}\circ\mathrm{pr}^{d+d,d}_{1}+\mathrm{pr}^{d+d,d}_{2})$ of the map $\tau$ of Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings §synchronous with the Borel map $\mathrm{pr}_{2}\circ\mathrm{pr}^{d+d,d}_{1}$.

$\kappa_{n}$ is a coupling of $S_{s_{n}}\mu_{n}$ and $\mu$. Push-forwards compose, $(T\circ S)_{\#}\rho=T_{\#}(S_{\#}\rho)$, directly from the definition in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. With $\tau$ the map of Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings §synchronous formed with $\gamma=\beta_{n}$ and time $s_{n}$, we have $\mathrm{pr}_{1}\circ\sigma=\mathrm{pr}_{1}\circ\tau$, so $(\mathrm{pr}_{1})_{\#}\kappa_{n}=(\mathrm{pr}_{1})_{\#}(\tau_{\#}P_{n})=(\mathrm{pr}_{1})_{\#}\beta_{n}^{s_{n}}=S_{s_{n}}\mu_{n}$, the last equality because $\beta_{n}^{s_{n}}\in\Pi(S_{s_{n}}\mu_{n},S_{s_{n}}\mu)$ by that clause. Also $\mathrm{pr}_{2}\circ\sigma=\mathrm{pr}_{2}\circ\mathrm{pr}^{d+d,d}_{1}$, and $(\mathrm{pr}^{d+d,d}_{1})_{\#}P_{n}=\beta_{n}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, so $(\mathrm{pr}_{2})_{\#}\kappa_{n}=(\mathrm{pr}_{2})_{\#}\beta_{n}=\mu$. By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling, $\kappa_{n}\in\Pi(S_{s_{n}}\mu_{n},\mu)$.

Cost of $\kappa_{n}$. For $a,b\in\mathbb{R}^{d}$, $\pi(a)-a\in\mathbb{Z}^{d}$ by The Half-Open Unit Cell Tiles Euclidean Space §wrap, so $d_{\mathbb{T}}(\pi(a),b)=d_{\mathbb{T}}(a,b)$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §symmetry. Hence, for $x,x',z\in\mathbb{R}^{d}$, by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §triangle, The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §symmetry and The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §minimal,

$$d_{\mathbb{T}}(\pi(x+z),x')\le d_{\mathbb{T}}(x+z,x)+d_{\mathbb{T}}(x,x')\le\lVert z\rVert+d_{\mathbb{T}}(x,x'),\qquad d_{\mathbb{T}}(\pi(x+z),x')^{2}\le2\lVert z\rVert^{2}+2\,d_{\mathbb{T}}(x,x')^{2}.$$

Integrating against $P_{n}$ and using the change-of-variables formula, the marginals $(\mathrm{pr}^{d+d,d}_{1})_{\#}P_{n}=\beta_{n}$ and $(\mathrm{pr}^{d+d,d}_{2})_{\#}P_{n}=N_{n}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, claim 3 of Image Measures, Measures with Densities, and Change of Variables for the density $g_{s_{n}}$ of $N_{n}$, and The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §moments with $q=d$ (note $s_{n}\le1$), we get

$$I_{\mathbb{T}}(\kappa_{n})\le2\int\lVert z\rVert^{2}\,N_{n}(dz)+2\,I_{\mathbb{T}}(\beta_{n})=2d\,s_{n}+2\,W_{\mathbb{T}}(\mu_{n},\mu)^{2}\longrightarrow0,$$

since $s_{n}\to0$ and $W_{\mathbb{T}}(\mu_{n},\mu)\to0$ by Convergent Sequence in a Metric Space. Also $I_{\mathbb{T}}(\beta_{n})=W_{\mathbb{T}}(\mu_{n},\mu)^{2}\to0$.

First sequence. The measures $S_{s_{n}}\mu_{n}$ and $\mu$ lie in $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, $\kappa_{n}\in\Pi(S_{s_{n}}\mu_{n},\mu)$ and $I_{\mathbb{T}}(\kappa_{n})\to0$, so condition (c) of intrinsic test functions gives $D_{\kappa_{n}}(F_{n},G)\to0$. By Convergence Along Couplings of Measures Carrying Square-Integrable Vector Fields on the Torus §convergence, with the couplings $\kappa_{n}$, the sequence $((S_{s_{n}}\mu_{n},\nabla\varphi(S_{s_{n}}\mu_{n})))$ converges along couplings to $(\mu,\nabla\varphi(\mu))$.

Second sequence. Let $w'\in\mathbb{R}^{d+d}$ with $x=\mathrm{pr}_{1}(w')$, $x'=\mathrm{pr}_{2}(w')$. By Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings §average (at time $s_{n}$), $a_{n}(x)=\int F_{n}(\pi(x+z))\,N_{n}(dz)$, so, $N_{n}$ being a probability measure, $a_{n}(x)-G(x')=\int\bigl(F_{n}(\pi(x+z))-G(x')\bigr)N_{n}(dz)$. Applying Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),N_{n})$ to each component and the constant $1$, and summing over the components (the inequality is trivial if the right side is infinite),

$$\lVert a_{n}(x)-G(x')\rVert^{2}\le\int\bigl\lVert F_{n}(\pi(x+z))-G(x')\bigr\rVert^{2}N_{n}(dz).$$

The integrand on the right is $H(\iota^{d+d,d}(w',z))$ with $H(v)=\lVert F_{n}(\mathrm{pr}_{1}(\sigma(v)))-G(\mathrm{pr}_{2}(\sigma(v)))\rVert^{2}$, a Borel $[0,\infty]$-valued map. Integrating against $\beta_{n}$ and using Tonelli's theorem (Tonelli and Fubini Theorems) together with the identity $\int H\,dP_{n}=\int H\circ\iota\,d(\beta_{n}\otimes N_{n})$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, then the change-of-variables formula for $\kappa_{n}=\sigma_{\#}P_{n}$,

$$D_{\beta_{n}}(a_{n},G)\le\int H\,dP_{n}=\int\lVert F_{n}(\mathrm{pr}_{1}(v'))-G(\mathrm{pr}_{2}(v'))\rVert^{2}\,\kappa_{n}(dv')=D_{\kappa_{n}}(F_{n},G)\longrightarrow0 .$$

Since also $I_{\mathbb{T}}(\beta_{n})\to0$ and $\beta_{n}\in\Pi(\mu_{n},\mu)$, Convergence Along Couplings of Measures Carrying Square-Integrable Vector Fields on the Torus §convergence shows that $((\mu_{n},A^{\mu_{n}}_{s_{n}}\nabla\varphi(S_{s_{n}}\mu_{n})))$ converges along couplings to $(\mu,\nabla\varphi(\mu))$. $\blacksquare$