Proof of The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property

Each result cited below is universally quantified over the data in its own statement. Integrals of nonnegative Borel functions are taken in $[0,\infty]$; 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, and change of variables under a push-forward, including the transfer of integrability, is that of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. Finite entropy, $\mathrm{Ent}$ and $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$, and $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ of Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §finite, are as in those definitions; $c_{\min}$ is the least variance of The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances and $Z_{c}$ the constant of The Diagonal Gaussian Density on Euclidean Space and Its Notation §density. By The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair, $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$.

Step 0 (Preliminaries). (a) The penalty domain. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. 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, $\mu$ has finite relative entropy with respect to $\gamma_{c}$ if and only if it has finite entropy. Hence $\mathcal{D}=\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ as sets, and for $\mu\in\mathcal{D}$, by the same clause and The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair,

$$\mathcal{E}(\mu)=a\,\mathrm{Ent}(\mu)+\tfrac{a}{2}\int_{\mathbb{R}^{d}}|x|_{c}^{2}\,\mu(dx)+a\,Z_{c},\tag{0.1}$$

the function $x\mapsto|x|_{c}^{2}$ being $\mu$-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.

(b) The score domain. 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 §score, a $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ has finite Fisher information relative to $\gamma_{c}$ if and only if $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$. With (a) and The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair, $\mathcal{D}_{\Sigma}=\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})\cap\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$.

(c) The quadratic form. Let $M_{c}$ be the real $d\times d$ matrix with diagonal entries $1/c_{1},\dots,1/c_{d}$ and all other entries $0$. It equals its transpose, so $M_{c}\in\mathcal{S}(d)$, and the $i$th entry of $M_{c}z$ is $z_{i}/c_{i}$, so $M_{c}z=S_{c}(z)$ and $z\cdot(M_{c}z)=|z|_{c}^{2}$ for $z\in\mathbb{R}^{d}$ (The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling). By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic (with $n=d$, the matrix $2M_{c}$, linear coefficient $0_{\mathbb{R}^{d}}$ and constant $0$) the function $z\mapsto|z|_{c}^{2}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ (an open subset of itself by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), hence continuous as a map into $\mathbb{R}$ with the absolute-value metric by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, hence Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps). For $x,y\in\mathbb{R}^{d}$, expanding $(x_{i}+y_{i})^{2}/c_{i}$ and summing over $i\in[d]$ gives

$$|x+y|_{c}^{2}=|x|_{c}^{2}+2\,S_{c}(x)\cdot y+|y|_{c}^{2},\qquad S_{c}(x)\cdot y=\sum_{i=1}^{d}\frac{x_{i}y_{i}}{c_{i}}=S_{c}(y)\cdot x .\tag{0.2}$$

Since $0<c_{\min}\le c_{i}$, we have $x_{i}^{2}/c_{i}\le c_{\min}^{-1}x_{i}^{2}$, and summing with Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square, $0\le|x|_{c}^{2}\le c_{\min}^{-1}\lVert x\rVert^{2}$.

Step S (Smoothing). Let $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $s\in\mathbb{R}$ with $0<s\le\tfrac12$, and let $\rho_{s}$ be the measure written $\mu_{s}$ in Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability built from $\rho$ and $s$. By Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability §density, $\rho_{s}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $W_{2}(\rho_{s},\rho)^{2}\le d\,s$; by Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability §entropy and Gaussian Smoothing of a Measure with Finite Second Moment: Positive Density, Finite Entropy, a Tangent Score, Non-Increasing Fisher Information and Exponential Integrability §score, $\rho_{s}\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})\cap\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$. By Step 0(b), $\rho_{s}\in\mathcal{D}_{\Sigma}$.

Step 1 (Clause 4: lower bound). Let $\mu\in\mathcal{D}$. The Gibbs inequality Relative Entropy on Euclidean Space: the Gibbs Inequality, a Variational Criterion for Finite Relative Entropy, and Closed Sublevel Sets under Weak Convergence §gibbs, with $m=d$, $\gamma=\gamma_{c}\in\mathcal{P}(\mathbb{R}^{d})$ (Diagonal Gaussian Measures on Euclidean Space §measure) and $\nu=\mu$, gives $0\le H(\mu\,|\,\gamma_{c})$; as $a>0$, $0\le a\,H(\mu\,|\,\gamma_{c})=\mathcal{E}(\mu)$. This is clause 4.

Step 2 (Clause 5: moment bound). Let $\mu\in\mathcal{D}$. By The Entropy Inequality for Nonnegative Functions, and Quadratic Moments Bounded by the Relative Entropy with Respect to a Diagonal Gaussian Measure §second-moment, $M_{2}(\mu)\le4c_{\max}H(\mu\,|\,\gamma_{c})+2\sum_{i=1}^{d}c_{i}$, and $H(\mu\,|\,\gamma_{c})=a^{-1}\mathcal{E}(\mu)$ by The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair, so $M_{2}(\mu)\le\frac{4c_{\max}}{a}\mathcal{E}(\mu)+2\sum_{i=1}^{d}c_{i}$. This is clause 5.

Step 3 (Clause 2: translations). Let $\mu\in\mathcal{D}$ and $h\in\mathbb{R}^{d}$. Each component $x\mapsto x_{i}+h_{i}$ of $\tau_{h}$ is continuous, since $|(x_{i}+h_{i})-(x'_{i}+h_{i})|=|x_{i}-x'_{i}|\le\lVert x-x'\rVert$ for all $x,x'\in\mathbb{R}^{d}$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate applied to $x-x'$, so $\tau_{h}$ is Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps). By Step 0(a), $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$, so 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 §translation gives $(\tau_{h})_{\#}\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})=\mathcal{D}$ and $\mathrm{Ent}((\tau_{h})_{\#}\mu)=\mathrm{Ent}(\mu)$. By The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean, each coordinate $x\mapsto x_{i}$ is Borel and $\mu$-integrable with integral $m(\mu)_{i}$; so by linearity the function $x\mapsto2\,S_{c}(h)\cdot x=\sum_{i=1}^{d}(2h_{i}/c_{i})x_{i}$ is $\mu$-integrable with integral $2\,S_{c}(h)\cdot m(\mu)$. By (0.2) with $y=h$, $|x+h|_{c}^{2}=|x|_{c}^{2}+2\,S_{c}(h)\cdot x+|h|_{c}^{2}$; the right side is $\mu$-integrable by Step 0(a) and linearity (the constant $|h|_{c}^{2}$ having integral $|h|_{c}^{2}$, $\mu$ being a probability measure). The nonnegative Borel function $|\cdot|_{c}^{2}$ therefore satisfies, by change of variables,

$$\int_{\mathbb{R}^{d}}|y|_{c}^{2}\,\bigl((\tau_{h})_{\#}\mu\bigr)(dy)=\int_{\mathbb{R}^{d}}|x+h|_{c}^{2}\,\mu(dx)=\int_{\mathbb{R}^{d}}|x|_{c}^{2}\,\mu(dx)+2\,S_{c}(h)\cdot m(\mu)+|h|_{c}^{2}.$$

Applying (0.1) to $(\tau_{h})_{\#}\mu$ and to $\mu$,

$$\mathcal{E}\bigl((\tau_{h})_{\#}\mu\bigr)=a\,\mathrm{Ent}(\mu)+\tfrac{a}{2}\int_{\mathbb{R}^{d}}|x|_{c}^{2}\,\mu(dx)+a\,Z_{c}+a\,S_{c}(h)\cdot m(\mu)+\tfrac{a}{2}|h|_{c}^{2}=\mathcal{E}(\mu)+a\,S_{c}(h)\cdot m(\mu)+\tfrac{a}{2}|h|_{c}^{2}.$$

This is the displayed identity of clause 2. By (0.2), $S_{c}(h)\cdot m(\mu)=S_{c}(m(\mu))\cdot h$, and by Step 0(c), $\tfrac{a}{2}|h|_{c}^{2}=\tfrac12\,h\cdot\bigl((aM_{c})h\bigr)$. Hence the function $e_{\mu}(h)=\mathcal{E}((\tau_{h})_{\#}\mu)$ of Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §translation is

$$e_{\mu}(h)=\tfrac12\,h\cdot\bigl((aM_{c})h\bigr)+\bigl(a\,S_{c}(m(\mu))\bigr)\cdot h+\mathcal{E}(\mu)\qquad(h\in\mathbb{R}^{d}),$$

and Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, with $n=d$, the matrix $aM_{c}$ (diagonal, its off-diagonal entries being $a\cdot0=0$, hence equal to its transpose and so in $\mathcal{S}(d)$), linear coefficient $a\,S_{c}(m(\mu))$ and constant $\mathcal{E}(\mu)$, shows that $e_{\mu}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with $D^{2}e_{\mu}(z)=aM_{c}$ for every $z$. These are the notions of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives (clause 3 of C^k Maps on a Euclidean Open Set and Hessian Matrix of a C^2 Function). Once the pair is known to be a penalty pair (Step 7), Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §hessian gives $H_{\mathcal{E}}(\mu)=D^{2}e_{\mu}(0_{\mathbb{R}^{d}})=aM_{c}$, the diagonal matrix with $i$th diagonal entry $a/c_{i}$. This completes clause 2.

Step 4 (Clause 3: first variation). Let $\mu\in\mathcal{D}$ and $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. A bound on second derivatives. For $i\in[d]$, $\partial_{i}\psi$ is smooth by claim 3 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map and, $\psi$ being of class $C^{1}$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient) and compactly supported, $\partial_{i}\psi$ is compactly supported by Integration by Parts on Euclidean Space Against a Compactly Supported Function of Class $C^{1}$ §vanishing with $h=\psi$; applying that clause again with $h=\partial_{i}\psi$, which is of class $C^{1}$ because it is smooth (Smooth Map on a Euclidean Open Set, $\psi$ being smooth by Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space), each $\partial_{j}\partial_{i}\psi$ ($j\in[d]$) is continuous and compactly supported, hence bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable (the Euclidean distance on $\mathbb{R}$ being the absolute-value metric by The Euclidean Distance on the Real Line is the Absolute Value Metric). Choose bounds $B_{ij}\ge0$ with $|\partial_{j}\partial_{i}\psi|\le B_{ij}$ and put $b=\sum_{i,j\in[d]}B_{ij}$, a nonnegative real number; as every summand is nonnegative, $B_{ij}\le b$, so $|\partial_{j}\partial_{i}\psi(x)|\le B_{ij}\le b$ for all $x$ and $i,j$.

The entropy part. By Step 0(a), $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$. By The First Variation of the Entropy Along a Gradient Perturbation of the Identity §variation, applied with $\mu$, $\psi$ and $b$, there is a positive $t_{0}$ such that $\nu_{t}=(\mathrm{id}+t\nabla\psi)_{\#}\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})=\mathcal{D}$ for every $t\in(-t_{0},t_{0})$, and $g(t)=\mathrm{Ent}(\nu_{t})$ defines a function on $(-t_{0},t_{0})$ differentiable at $0$ with derivative $-\int\Delta\psi\,d\mu$. The map $\mathrm{id}+t\nabla\psi$ is Borel by 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.

The quadratic part. By Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test, $\nabla\psi$ is Borel with $\int\lVert\nabla\psi\rVert^{2}\,d\mu<\infty$; by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient there is $K_{\psi}\ge0$ with $\lVert\nabla\psi\rVert\le K_{\psi}$. The function $x\mapsto|\nabla\psi(x)|_{c}^{2}$ is Borel (a composition of Borel maps, Step 0(c)) and bounded by $c_{\min}^{-1}K_{\psi}^{2}$ (Step 0(c)), hence $\mu$-integrable (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures); put $K=\int|\nabla\psi|_{c}^{2}\,d\mu\ge0$. The classes $S_{c}$ (Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information) and $\nabla\psi$ lie in $L^{2}(\mu;\mathbb{R}^{d})$, so by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu the function $x\mapsto S_{c}(x)\cdot\nabla\psi(x)$ is $\mu$-integrable with integral $B=\langle S_{c},\nabla\psi\rangle_{\mu}$. Put $A=\int|x|_{c}^{2}\,\mu(dx)$. For every $t\in\mathbb{R}$ and $x$, (0.2) with $y=t\nabla\psi(x)$ gives $|x+t\nabla\psi(x)|_{c}^{2}=|x|_{c}^{2}+2t\,S_{c}(x)\cdot\nabla\psi(x)+t^{2}|\nabla\psi(x)|_{c}^{2}$, a $\mu$-integrable function with integral $A+2tB+t^{2}K$ by linearity. By change of variables for the nonnegative Borel function $|\cdot|_{c}^{2}$,

$$F(t)=\int_{\mathbb{R}^{d}}|y|_{c}^{2}\,\nu_{t}(dy)=\int_{\mathbb{R}^{d}}|x+t\nabla\psi(x)|_{c}^{2}\,\mu(dx)=A+2tB+t^{2}K\qquad(t\in(-t_{0},t_{0})).$$

The function $F$ on $(-t_{0},t_{0})$ is differentiable at $0$ with derivative $2B$ in the sense of Derivative at an Interior Point: $0$ is an interior point of the interval $(-t_{0},t_{0})$ by Basic Facts about Intervals of the Real Line and Their Interior Points §open-interval, and given $\varepsilon>0$ put $\delta=\varepsilon/(K+1)>0$; for $0<|t|<\delta$ with $t\in(-t_{0},t_{0})$, $\bigl|\tfrac{F(t)-F(0)}{t}-2B\bigr|=K|t|\le(K+1)|t|<\varepsilon$.

Conclusion. By (0.1) applied to $\nu_{t}\in\mathcal{D}$, $\mathcal{E}(\nu_{t})=a\,g(t)+\tfrac{a}{2}F(t)+a\,Z_{c}$ for $t\in(-t_{0},t_{0})$. By claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives on the interval $(-t_{0},t_{0})$, this function is differentiable at $0$ with derivative

$$-a\int_{\mathbb{R}^{d}}\Delta\psi\,d\mu+\tfrac{a}{2}\cdot2B=-a\Bigl(\int_{\mathbb{R}^{d}}\Delta\psi\,d\mu-\langle S_{c},\nabla\psi\rangle_{\mu}\Bigr)=-a\,\ell^{c}_{\mu}(\psi),$$

by Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §functional. This is clause 3.

Step 5 (Clause 6: lower semicontinuity). By Step 0(a), $\mathcal{D}=\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$, on which $\mathrm{Ent}$ is lower semicontinuous by 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 §lsc. We verify the condition of claim 3 of Sequential Characterization of Lower Semicontinuity on a Subset of a Metric Space for $u=\mathcal{E}$ on $A=\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. Let $\mu\in\mathcal{D}$, let $(\mu_{j})_{j}$ be a sequence in $\mathcal{D}$ converging to $\mu$ in $\mathcal{D}$ with the restricted metric, so that $\lim_{j}W_{2}(\mu_{j},\mu)=0$, and let $\varepsilon>0$. By claim 3 of Sequential Characterization of Lower Semicontinuity on a Subset of a Metric Space applied to $u=\mathrm{Ent}$ on $A=\mathcal{D}$, the condition of its claim 1 holds at $\mu$; applied to $(\mu_{j})_{j}$ and $\varepsilon/(2a)$, it gives $N_{1}$ with $\mathrm{Ent}(\mu)-\tfrac{\varepsilon}{2a}<\mathrm{Ent}(\mu_{j})$ for $j\ge N_{1}$. The function $w(x)=|x|_{c}^{2}$ is continuous (Step 0(c)) with $|w(x)|\le c_{\min}^{-1}(1+\lVert x\rVert^{2})$, so by Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth §quadratic (with $m=d$) $\lim_{j}\int w\,d\mu_{j}=\int w\,d\mu$, and by Limit of a Sequence of Real Numbers there is $N_{2}$ with $\bigl|\int w\,d\mu_{j}-\int w\,d\mu\bigr|<\varepsilon/a$ for $j\ge N_{2}$. For $j$ at least the larger of $N_{1},N_{2}$, (0.1) gives

$$\mathcal{E}(\mu_{j})-\mathcal{E}(\mu)=a\bigl(\mathrm{Ent}(\mu_{j})-\mathrm{Ent}(\mu)\bigr)+\tfrac{a}{2}\Bigl(\int w\,d\mu_{j}-\int w\,d\mu\Bigr)>-\tfrac{\varepsilon}{2}-\tfrac{\varepsilon}{2}=-\varepsilon .$$

So $\mathcal{E}(\mu)-\varepsilon<\mathcal{E}(\mu_{j})$ for all such $j$, and claim 3 of Sequential Characterization of Lower Semicontinuity on a Subset of a Metric Space shows that $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$. This is clause 6.

Step 6 (Clause 7: map property). Every $\mu\in\mathcal{D}$ has finite entropy (Step 0(a)) and is therefore absolutely continuous by 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. As $\mathcal{D}\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$, Two Sufficient Conditions for the Map Property: Absolute Continuity, and Atomlessness on the Line §absolutely-continuous with $Q=\mathcal{D}$ shows that $\mathcal{D}$ has the map property of The Map Property of a Set of Probability Measures §map-property, which is the map property of The Intrinsic Calculus on the Wasserstein Space: Standing Notation §maps. This is clause 7.

Step 7 (Clause 1: penalty pair). The data have the form required by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair: $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$, $\mathcal{E}(\mu)=a\,H(\mu\,|\,\gamma_{c})$ is a real number for $\mu\in\mathcal{D}$, and for $\mu\in\mathcal{D}_{\Sigma}$ the relative score $\zeta^{c}_{\mu}$ lies in $T_{\mu}$ by that clause, and $T_{\mu}$ is a linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$ (Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed), so $\Sigma(\mu)=a\,\zeta^{c}_{\mu}\in T_{\mu}$. We verify the five conditions.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty. $\gamma_{c}$ is a probability measure on $\mathbb{R}^{d}$ (Diagonal Gaussian Measures on Euclidean Space §measure); the constant function $1$ is a density of $\gamma_{c}$ with respect to itself, as $\int_{B}1\,d\gamma_{c}=\gamma_{c}(B)$ for every Borel $B$, and $\phi(1)=1\cdot\log1=0$ for $\phi(s)=s\log s$, so $\phi\circ1=0$ is $\gamma_{c}$-integrable and $\gamma_{c}$ has finite relative entropy with respect to itself (Relative Entropy of Probability Measures §relative-entropy). Hence $\gamma_{c}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Entropy Inequality for Nonnegative Functions, and Quadratic Moments Bounded by the Relative Entropy with Respect to a Diagonal Gaussian Measure §second-moment, and Step S with $\rho=\gamma_{c}$ and $s=\tfrac12$ gives an element of $\mathcal{D}_{\Sigma}$.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §bound. Take $C=0$: by Step 1, $-0\cdot(1+M_{2}(\mu))=0\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §translation. By Step 3, $(\tau_{h})_{\#}\mu\in\mathcal{D}$ and $e_{\mu}$ is of class $C^{2}$ on $\mathbb{R}^{d}$, for every $\mu\in\mathcal{D}$ and $h\in\mathbb{R}^{d}$.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §variation. Let $\mu\in\mathcal{D}_{\Sigma}$ and $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. Step 4 gives $t_{0}>0$ with $(\mathrm{id}+t\nabla\psi)_{\#}\mu\in\mathcal{D}$ for $t\in(-t_{0},t_{0})$ and $t\mapsto\mathcal{E}((\mathrm{id}+t\nabla\psi)_{\#}\mu)$ differentiable at $0$ with derivative $-a\,\ell^{c}_{\mu}(\psi)$. By Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §score, $-\ell^{c}_{\mu}(\psi)=\langle\zeta^{c}_{\mu},\nabla\psi\rangle_{\mu}$, so by bilinearity of the inner product the derivative is $\langle a\,\zeta^{c}_{\mu},\nabla\psi\rangle_{\mu}=\langle\Sigma(\mu),\nabla\psi\rangle_{\mu}$.

Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §dense. Let $\mu\in\mathcal{D}$ and $\varepsilon>0$. Let $s$ be the lesser of $\tfrac12$ and $\varepsilon^{2}/(2d)$, with $d$ read in $\mathbb{R}$; then $0<s\le\tfrac12$, and Step S with $\rho=\mu$ gives $\mu_{s}\in\mathcal{D}_{\Sigma}$ with $W_{2}(\mu_{s},\mu)^{2}\le d\,s\le\varepsilon^{2}/2<\varepsilon^{2}$. Both $W_{2}(\mu_{s},\mu)$ and $\varepsilon$ being nonnegative, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $W_{2}(\mu_{s},\mu)<\varepsilon$.

Hence $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, which is clause 1; in particular the translation Hessian $H_{\mathcal{E}}(\mu)$ is defined, and Step 3 identifies it. $\blacksquare$