Proof of The Gaussian Free-Energy Pair is Uniformly Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named at the point of use. Elementary arithmetic and order facts for real numbers (The Real Numbers: Standing Notation and Background §background) are used without citation. Linearity and monotonicity of integrals are Linearity and Monotonicity of the Lebesgue Integral §nonnegative for nonnegative measurable functions and Linearity and Monotonicity of the Lebesgue Integral §integrable for integrable ones. Finite entropy, the entropy $\mathrm{Ent}$ and the set $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ are those of that definition; $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ is the set of measures of finite Fisher information and $\xi_{\rho}$ the score of $\rho\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$; finite relative entropy and $H(\cdot\,|\,\cdot)$ are those of Relative Entropy of Probability Measures §relative-entropy; and $T_{\rho}$ is the tangent space at $\rho$. For $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the inner product of $L^{2}(\rho;\mathbb{R}^{d})$ is $\langle f,g\rangle_{\rho}=\int f\cdot g\,d\rho$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu). For $z\in\mathbb{R}^{d+d}$ we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$ (The Intrinsic Calculus on the Wasserstein Space: Standing Notation §couplings); if $\pi\in\Pi(\rho,\rho')$ and $f$ is Borel, then $\int f(x)\,\pi(dz)=\int f\,d\rho$ and $\int f(y)\,\pi(dz)=\int f\,d\rho'$ whenever either side is defined (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward). Write $S_{c}$, $|\cdot|_{c}^{2}$, $Z_{c}$ and $c_{\min}$ for the scaling map, the weighted square, the normalizing constant and the least variance of $c$, and $\gamma_{c}$ for the diagonal Gaussian measure with variances $c$. By The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair and The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §pair, the quadruple is a penalty pair, $\mathcal{D}$ is the set of $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ of finite relative entropy with respect to $\gamma_{c}$, with $\mathcal{E}(\rho)=a\,H(\rho\,|\,\gamma_{c})$, and $\mathcal{D}_{\Sigma}$ is the set of $\rho\in\mathcal{D}$ of finite Fisher information relative to $\gamma_{c}$, with $\Sigma(\rho)=a\,\zeta^{c}_{\rho}$.

Step 0 (the relative score against test functions and test fields). Let $\rho\in\mathcal{D}_{\Sigma}$. By Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §score, for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$,

$$\langle\Sigma(\rho),\nabla\psi\rangle_{\rho}=-a\,\ell^{c}_{\rho}(\psi)=-a\int\Delta\psi\,d\rho+a\int S_{c}\cdot\nabla\psi\,d\rho ,$$

the second equality being the formula of Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §functional. Moreover, as $\rho$ has finite Fisher information relative to $\gamma_{c}$, Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure: Comparison with the Entropy, the Score and the Relative Free Energy of a Quadratic Potential §score gives $\rho\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and $\zeta^{c}_{\rho}=\xi_{\rho}+S_{c}$ in $T_{\rho}$. Let $\eta$ be a test field. Its components are smooth and compactly supported, hence of class $C^{1}$, so The Score Integrates by Parts Against Every Compactly Supported Continuously Differentiable Vector Field §parts gives $\langle\xi_{\rho},\eta\rangle_{\rho}=-\int\operatorname{div}\eta\,d\rho$, and by bilinearity

$$\langle\Sigma(\rho),\eta\rangle_{\rho}=a\langle\xi_{\rho},\eta\rangle_{\rho}+a\langle S_{c},\eta\rangle_{\rho}=-a\int\operatorname{div}\eta\,d\rho+a\int S_{c}\cdot\eta\,d\rho .$$

Step 1 (integrands of quadratic growth). By Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information, $S_{c}$ is continuous and $\lVert S_{c}(x)\rVert^{2}\le c_{\min}^{-2}\lVert x\rVert^{2}$, so $\lVert S_{c}(x)\rVert\le c_{\min}^{-1}\lVert x\rVert$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Its $l$-th component $x\mapsto x_{l}/c_{l}$ is continuous, since $|x_{l}/c_{l}-x'_{l}/c_{l}|\le\lVert S_{c}(x)-S_{c}(x')\rVert$ for all $x,x'$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate. Also $\lVert x\rVert\le1+\lVert x\rVert^{2}$ for every $x$ (if $\lVert x\rVert\le1$ this is clear, and otherwise $\lVert x\rVert\le\lVert x\rVert^{2}$). Now let $v:\mathbb{R}^{d}\to\mathbb{R}^{d}$ have continuous components $v_{l}$ and satisfy $\lVert v(x)\rVert\le B$ for all $x$, for some real $B\ge0$. Then $S_{c}\cdot v=\sum_{l}(x_{l}/c_{l})\,v_{l}$ is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set (applied repeatedly), and by Cauchy-Schwarz Inequality for the Euclidean Dot Product,

$$|S_{c}(x)\cdot v(x)|\le\lVert S_{c}(x)\rVert\,\lVert v(x)\rVert\le c_{\min}^{-1}B\,\bigl(1+\lVert x\rVert^{2}\bigr).$$

Likewise a continuous $f:\mathbb{R}^{d}\to\mathbb{R}$ with $|f|\le B$ satisfies $|f(x)|\le B(1+\lVert x\rVert^{2})$. Hence, whenever $(\nu_{n})_{n}$ and $\nu$ lie in $\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\nu_{n},\nu)\to0$, Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth §quadratic gives $\int S_{c}\cdot v\,d\nu_{n}\to\int S_{c}\cdot v\,d\nu$ and $\int f\,d\nu_{n}\to\int f\,d\nu$. We apply this to $v=\nabla\psi$ and to $f\in\{\Delta\psi,\lVert\nabla\psi\rVert^{2}\}$ for $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$: the partial derivatives $\partial_{l}\psi$ are continuous and $\lVert\nabla\psi\rVert$ is bounded by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient, so $\lVert\nabla\psi\rVert^{2}=\sum_{l}(\partial_{l}\psi)^{2}$ is continuous (Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, applied repeatedly) and bounded, and $\Delta\psi$ is continuous and bounded by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §laplacian. We also apply it to a test field $v=\eta$ and to $f=\operatorname{div}\eta=\sum_{l}\partial_{l}\eta_{l}$: by Vector Fields with Test-Function Components are Dense in the Square-Integrable Vector Fields Against a Measure of Finite Second Moment the components $\eta_{l}$ are continuous and there is $B\ge0$ with $\lVert\eta(x)\rVert^{2}\le dB^{2}$, so $\lVert\eta(x)\rVert\le B_{\eta}$ with $B_{\eta}$ the nonnegative square root of $dB^{2}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field); and each $\partial_{l}\eta_{l}$ is continuous and bounded by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient applied to $\eta_{l}$, so $\operatorname{div}\eta$ is continuous (Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, applied repeatedly) and bounded.

Claim 1 (uniform displacement convexity). Let $\mu\in\mathcal{D}_{\Sigma}$, $\nu\in\mathcal{D}$, and let $\pi\in\Pi(\mu,\nu)$ be optimal (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions). By Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure: Comparison with the Entropy, the Score and the Relative Free Energy of a Quadratic Potential §entropy, $\mu$ and $\nu$ have finite entropy, hence lie in $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$, and

$$\mathcal{E}(\rho)=a\,\mathrm{Ent}(\rho)+\tfrac{a}{2}\int|x|_{c}^{2}\,\rho(dx)+a\,Z_{c}\qquad(\rho\in\{\mu,\nu\}),$$

the function $x\mapsto|x|_{c}^{2}$ being $\rho$-integrable by Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure: Comparison with the Entropy, the Score and the Relative Free Energy of a Quadratic Potential §integrable. By Step 0, $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and $\Sigma(\mu)=a\,\xi_{\mu}+a\,S_{c}$, so by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §linear, $\mathcal{J}(\Sigma(\mu),\pi)=a\,\mathcal{J}(\xi_{\mu},\pi)+a\,\mathcal{J}(S_{c},\pi)$.

Entropy. As $\mu$ has finite entropy it is absolutely continuous (Basic Properties of the Entropy on the Wasserstein Space: Comparison with the Gaussian Relative Entropy, Lower Bound, Translation Invariance, Absolute Continuity, Closed Sublevel Sets and Lower Semicontinuity §absolutely-continuous), so by Brenier's Theorem: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map §map there is a Borel $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\pi=(\mathrm{id},T)_{\#}\mu$ and $T_{\#}\mu=\nu$; thus $T$ is an optimal map from $\mu$ to $\nu$. 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 §pushforward and 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, $\int\lVert T-\mathrm{id}\rVert^{2}\,d\mu=I(\pi)<\infty$, so The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement with $S=T$ (for which $\pi_{S}=\pi$) gives $\mathcal{J}(\xi_{\mu},\pi)=\langle\xi_{\mu},T-\mathrm{id}\rangle_{\mu}$. Hence McCann's Tangent Inequality: the Entropy Lies Above its Tangent Along Optimal Maps §tangent gives

$$\mathrm{Ent}(\mu)+\mathcal{J}(\xi_{\mu},\pi)\le\mathrm{Ent}(\nu).$$

Quadratic part. For $x,y\in\mathbb{R}^{d}$, by The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling,

$$\tfrac12|y|_{c}^{2}-\tfrac12|x|_{c}^{2}-S_{c}(x)\cdot(y-x)=\tfrac12\sum_{i=1}^{d}\frac{y_{i}^{2}-x_{i}^{2}-2x_{i}y_{i}+2x_{i}^{2}}{c_{i}}=\tfrac12\sum_{i=1}^{d}\frac{(y_{i}-x_{i})^{2}}{c_{i}}\ge\frac{1}{2c_{\max}}\sum_{i=1}^{d}(y_{i}-x_{i})^{2}=\frac{1}{2c_{\max}}\lVert y-x\rVert^{2},$$

since $0<c_{i}\le c_{\max}$ (The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances) and by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square. The functions $z\mapsto|x|_{c}^{2}$ and $z\mapsto|y|_{c}^{2}$ are $\pi$-integrable with integrals $\int|x|_{c}^{2}\,\mu(dx)$ and $\int|y|_{c}^{2}\,\nu(dy)$; by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §pairing with the representative $S_{c}$, the function $z\mapsto S_{c}(x)\cdot(y-x)$ is $\pi$-integrable with integral $\mathcal{J}(S_{c},\pi)$; and $\int\lVert x-y\rVert^{2}\,\pi(dz)=I(\pi)$ is finite (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost, Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite). Integrating the pointwise inequality against $\pi$,

$$\tfrac12\int|x|_{c}^{2}\,\mu(dx)+\mathcal{J}(S_{c},\pi)+\frac{1}{2c_{\max}}\,I(\pi)\le\tfrac12\int|y|_{c}^{2}\,\nu(dy).$$

Combination. Put $\lambda=a/c_{\max}$, so that $\tfrac{\lambda}{2}I(\pi)=a\cdot\frac{1}{2c_{\max}}I(\pi)$. Multiplying the entropy inequality and the quadratic inequality by $a>0$ and adding $a\,Z_{c}$,

$$\mathcal{E}(\mu)+\mathcal{J}(\Sigma(\mu),\pi)+\tfrac{\lambda}{2}I(\pi)=a\bigl(\mathrm{Ent}(\mu)+\mathcal{J}(\xi_{\mu},\pi)\bigr)+a\Bigl(\tfrac12\int|x|_{c}^{2}\,d\mu+\mathcal{J}(S_{c},\pi)+\tfrac{1}{2c_{\max}}I(\pi)\Bigr)+a\,Z_{c}\le a\,\mathrm{Ent}(\nu)+\tfrac{a}{2}\int|y|_{c}^{2}\,d\nu+a\,Z_{c}=\mathcal{E}(\nu).$$

So the pair is $\frac{a}{c_{\max}}$-displacement convex ($\lambda$-Displacement Convexity of a Penalty Pair on the Wasserstein Space §convex). Since $I(\pi)\ge0$ and $\frac{a}{c_{\max}}>0$, also $\mathcal{E}(\mu)+\mathcal{J}(\Sigma(\mu),\pi)+\tfrac{0}{2}I(\pi)\le\mathcal{E}(\nu)$, i.e. the pair is displacement convex.

Claim 2 (closed score). Let $R\ge0$, let $(\nu_{n})_{n}$ be a sequence in $\mathcal{D}_{\Sigma}$ with $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le R$, let $\nu\in\mathcal{D}$, and let $(\pi_{n})_{n}$ be a sequence of couplings of vanishing cost from $(\nu_{n})_{n}$ to $\nu$. Since $W_{2}(\nu_{n},\nu)^{2}\le I(\pi_{n})$ (The Quadratic Wasserstein Distance on Euclidean Space §distance) and $I(\pi_{n})\to0$, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $W_{2}(\nu_{n},\nu)\to0$, so Step 1 applies.

(a) $\nu\in\mathcal{D}_{\Sigma}$. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and put $\ell_{n}=\langle\Sigma(\nu_{n}),\nabla\psi\rangle_{\nu_{n}}=-a\,\ell^{c}_{\nu_{n}}(\psi)$ (Step 0). By the formula of Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §functional, Step 1, Arithmetic of Limits of Real Sequences §sums and Arithmetic of Limits of Real Sequences §scalar, $\ell_{n}\to-a\,\ell^{c}_{\nu}(\psi)$, and $\lVert\nabla\psi\rVert_{\nu_{n}}^{2}\to\lVert\nabla\psi\rVert_{\nu}^{2}$. By The Cauchy-Schwarz Inequality in a Real Inner Product Space in $L^{2}(\nu_{n};\mathbb{R}^{d})$, $\ell_{n}^{2}\le R^{2}\lVert\nabla\psi\rVert_{\nu_{n}}^{2}$; passing to the limit (Arithmetic of Limits of Real Sequences §products and Arithmetic of Limits of Real Sequences §scalar, claim 1 of Order Properties of Limits of Real Sequences), $(a\,\ell^{c}_{\nu}(\psi))^{2}\le(R\lVert\nabla\psi\rVert_{\nu})^{2}$, so $a\,|\ell^{c}_{\nu}(\psi)|\le R\lVert\nabla\psi\rVert_{\nu}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field) and $|\ell^{c}_{\nu}(\psi)|\le\frac{R}{a}\lVert\nabla\psi\rVert_{\nu}$. This holds for every $\psi$, so $\nu$ has finite Fisher information relative to $\gamma_{c}$ (Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §finite, with $C=R/a$); as $\nu\in\mathcal{D}$, $\nu\in\mathcal{D}_{\Sigma}$.

(b) Test fields. Let $\eta$ be a test field and $B_{\eta}$ as in Step 1. By Step 0 applied to $\rho=\nu_{n}$ and to $\rho=\nu$ (by (a)), and by Step 1 with Arithmetic of Limits of Real Sequences §sums and Arithmetic of Limits of Real Sequences §scalar,

$$\langle\Sigma(\nu_{n}),\eta\rangle_{\nu_{n}}=-a\int\operatorname{div}\eta\,d\nu_{n}+a\int S_{c}\cdot\eta\,d\nu_{n}\ \longrightarrow\ -a\int\operatorname{div}\eta\,d\nu+a\int S_{c}\cdot\eta\,d\nu=\langle\Sigma(\nu),\eta\rangle_{\nu}.$$

Next let $q_{n}$ be a Borel representative of $\Sigma(\nu_{n})$. By The Cross Pairing of Two Square-Integrable Vector Fields Along a Coupling §pairing, $\mathcal{K}(\Sigma(\nu_{n}),\eta,\pi_{n})=\int q_{n}(x)\cdot\eta(y)\,\pi_{n}(dz)$, while $\langle\Sigma(\nu_{n}),\eta\rangle_{\nu_{n}}=\int q_{n}(x)\cdot\eta(x)\,\pi_{n}(dz)$. The components of $\eta$ are continuous and compactly supported, hence uniformly continuous (A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Uniformly Continuous); given $\theta>0$ choose $\delta>0$ with $\lVert\eta(y)-\eta(x)\rVert^{2}<\theta$ whenever $\lVert x-y\rVert<\delta$ (take the least of the $d$ radii for the tolerance $(\theta/d)^{1/2}$ in each component and use Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square). Then for every $z$, $\lVert\eta(y)-\eta(x)\rVert^{2}\le\theta+4B_{\eta}^{2}\delta^{-2}\lVert x-y\rVert^{2}$, as $\lVert\eta(y)-\eta(x)\rVert\le2B_{\eta}$ (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §triangle, Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §homogeneity) and $1\le\delta^{-2}\lVert x-y\rVert^{2}$ when $\lVert x-y\rVert\ge\delta$. Integrating, $D_{n}=\int\lVert\eta(y)-\eta(x)\rVert^{2}\,\pi_{n}(dz)\le\theta+4B_{\eta}^{2}\delta^{-2}I(\pi_{n})$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost), which is below $2\theta$ for all large $n$; so $D_{n}\to0$. By Cauchy-Schwarz Inequality for the Euclidean Dot Product and Hoelder's Inequality, for Two and for Finitely Many Factors §holder with exponents $2$ and $2$,

$$\bigl|\mathcal{K}(\Sigma(\nu_{n}),\eta,\pi_{n})-\langle\Sigma(\nu_{n}),\eta\rangle_{\nu_{n}}\bigr|\le\int\lVert q_{n}(x)\rVert\,\lVert\eta(y)-\eta(x)\rVert\,\pi_{n}(dz)\le R\sqrt{D_{n}}\to0,$$

since $\int\lVert q_{n}(x)\rVert^{2}\,\pi_{n}(dz)=\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}^{2}\le R^{2}$. With claim 3 of Order Properties of Limits of Real Sequences, $\mathcal{K}(\Sigma(\nu_{n}),\eta,\pi_{n})\to\langle\Sigma(\nu),\eta\rangle_{\nu}$.

(c) All fields. Let $\eta_{0}\in L^{2}(\nu;\mathbb{R}^{d})$ and $\varepsilon>0$; put $S_{\nu}=\lVert\Sigma(\nu)\rVert_{\nu}$ and $\varepsilon'=\varepsilon/(3(R+S_{\nu}+1))$. By Vector Fields with Test-Function Components are Dense in the Square-Integrable Vector Fields Against a Measure of Finite Second Moment §dense there is a test field $\eta$ with $\lVert\eta_{0}-\eta\rVert_{\nu}\le\varepsilon'$. By The Cross Pairing of Two Square-Integrable Vector Fields Along a Coupling §linear and The Cross Pairing of Two Square-Integrable Vector Fields Along a Coupling §bound, $|\mathcal{K}(\Sigma(\nu_{n}),\eta_{0},\pi_{n})-\mathcal{K}(\Sigma(\nu_{n}),\eta,\pi_{n})|\le R\varepsilon'$ for every $n$, and by The Cauchy-Schwarz Inequality in a Real Inner Product Space, $|\langle\Sigma(\nu),\eta_{0}\rangle_{\nu}-\langle\Sigma(\nu),\eta\rangle_{\nu}|\le S_{\nu}\varepsilon'$. By (b) choose $N$ with $|\mathcal{K}(\Sigma(\nu_{n}),\eta,\pi_{n})-\langle\Sigma(\nu),\eta\rangle_{\nu}|<\varepsilon/3$ for $n\ge N$. For $n\ge N$, $|\mathcal{K}(\Sigma(\nu_{n}),\eta_{0},\pi_{n})-\langle\Sigma(\nu),\eta_{0}\rangle_{\nu}|<(R+S_{\nu})\varepsilon'+\varepsilon/3<\varepsilon$. So $(\Sigma(\nu_{n}))_{n}$ converges weakly to $\Sigma(\nu)$ along $(\pi_{n})_{n}$ (Strong and Weak Convergence of Vector Fields Along Couplings of Vanishing Cost §weak), and with (a) the pair has closed score along couplings (Penalty Pairs with Closed Score Along Couplings §closed).

Claim 3 (regular penalised maxima). Let $\chi$ be an intrinsic test function on $\mathcal{D}$, let $\lambda>0$, and let $\mu\in\mathcal{D}$ be a point at which $\chi-\lambda\mathcal{E}$ has a local maximum relative to $\mathcal{D}$, with radius $r_{0}>0$. By Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §differentiability, $\chi$ is differentiable along couplings at $\mu$ with gradient $w=\nabla\chi(\mu)\in L^{2}(\mu;\mathbb{R}^{d})$. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, $G_{t}=\mathrm{id}+t\nabla\psi$, let $t_{0}$ be as in The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §variation for $\mu$ and $\psi$, $p_{\psi}=\lVert\nabla\psi\rVert_{\mu}$, and $t_{2}$ the lesser of $t_{0}$ and $r_{0}/(p_{\psi}+1)$. For $t\in(-t_{2},t_{2})$: $(G_{t})_{\#}\mu\in\mathcal{D}$, and $W_{2}(\mu,(G_{t})_{\#}\mu)\le|t|p_{\psi}<r_{0}$ (The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound §distance); so $\phi(t)=\chi((G_{t})_{\#}\mu)-\lambda\mathcal{E}((G_{t})_{\#}\mu)\le\phi(0)$, as $(G_{0})_{\#}\mu=\mu$ (The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound §borel). Thus $\phi:(-t_{2},t_{2})\to\mathbb{R}$ has a local maximum at $0$.

Differentiability of the first term. Put $g_{0}=\langle w,\nabla\psi\rangle_{\mu}$. Let $\varepsilon>0$ and let $\theta>0$ be given by Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §differentiable for $\varepsilon/(p_{\psi}+1)$. For $t\in(-t_{2},t_{2})$ with $0<|t|<\theta/(p_{\psi}+1)$, the coupling $\pi_{t}=(\mathrm{id},G_{t})_{\#}\mu\in\Pi(\mu,(G_{t})_{\#}\mu)$ has $I(\pi_{t})=t^{2}p_{\psi}^{2}<\theta^{2}$ (The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound §coupling), and by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement with $S=G_{t}$, whose displacement $G_{t}-\mathrm{id}=t\nabla\psi$ is square-integrable against $\mu$ (The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient), $\mathcal{J}(w,\pi_{t})=\langle w,t\nabla\psi\rangle_{\mu}=tg_{0}$. Hence $|\chi((G_{t})_{\#}\mu)-\chi(\mu)-tg_{0}|\le\tfrac{\varepsilon}{p_{\psi}+1}|t|p_{\psi}<\varepsilon|t|$, so $t\mapsto\chi((G_{t})_{\#}\mu)$ on $(-t_{2},t_{2})$ is differentiable at $0$ with derivative $g_{0}$ (Derivative at an Interior Point).

Conclusion. By The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §variation, the function $t\mapsto\mathcal{E}((G_{t})_{\#}\mu)$ on $(-t_{0},t_{0})$ is differentiable at $0$ with derivative $-a\,\ell^{c}_{\mu}(\psi)$; by claim 2 of Restriction Stability of Continuity and of the Derivative so is its restriction to $(-t_{2},t_{2})$, and by claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\phi$ is differentiable at $0$ with $\phi'(0)=g_{0}+\lambda a\,\ell^{c}_{\mu}(\psi)$. By Vanishing of the Derivative at an Interior Local Extremum, $\phi'(0)=0$. With The Cauchy-Schwarz Inequality in a Real Inner Product Space in $L^{2}(\mu;\mathbb{R}^{d})$,

$$|\ell^{c}_{\mu}(\psi)|=(\lambda a)^{-1}|g_{0}|\le(\lambda a)^{-1}\lVert w\rVert_{\mu}\,\lVert\nabla\psi\rVert_{\mu}.$$

This holds for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, so $\mu$ has finite Fisher information relative to $\gamma_{c}$ (Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §finite, with $C=(\lambda a)^{-1}\lVert w\rVert_{\mu}$); as $\mu\in\mathcal{D}$, $\mu\in\mathcal{D}_{\Sigma}$. So the pair has regular penalised maxima (Penalty Pairs with Regular Penalised Maxima §regular). $\blacksquare$