Proof of The Gaussian Entropy Pair Has Regular Penalised Maxima

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

The quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair, so Noise Penalty Pairs with Regular Penalised Maxima §regular applies to it. Let $\chi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ be a noise intrinsic test function on $\mathcal{D}$, let $\lambda\in\mathbb{R}$ be positive, and let $\mu\in\mathcal{D}$ be a point at which $\chi-\lambda\mathcal{E}$ has a local maximum relative to $\mathcal{D}$ in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §space. By Local Maximum of a Function Relative to a Subset of a Metric Space there is a positive $r\in\mathbb{R}$ such that every $\nu\in\mathcal{D}$ with $W_{a}(\mu,\nu)<r$ satisfies $\chi(\nu)-\lambda\mathcal{E}(\nu)\le\chi(\mu)-\lambda\mathcal{E}(\mu)$; since $\mathcal{E}=\beta H(\cdot\,|\,\gamma_{c})$ on $\mathcal{D}$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair, this reads

$$\chi(\nu)-\chi(\mu)\le\lambda\beta\bigl(H(\nu\,|\,\gamma_{c})-H(\mu\,|\,\gamma_{c})\bigr).$$

We must show $\mu\in\mathcal{D}_{\Sigma}$. Since $\mu\in\mathcal{D}$, $\mu$ has finite relative entropy with respect to $\gamma_{c}$ and $\mu\in\mathcal{P}_{2}(X)$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain. By property (b) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §differentiability, $\chi$ is differentiable along noise couplings at $\mu$, and its gradient $\nabla\chi(\mu)\in L^{2}(\mu;X^{a})$ has the property of Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §differentiable by Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §gradient.

Step 1 (Data for a fixed cylindrical direction). Fix $n\in\mathbb{N}$ and $g\in C^{2}_{b}(\mathbb{R}^{n})$, and put $\psi=g\circ p_{n}$, a bounded $C^{1}$ cylindrical function by the preamble of Relative Entropy along a Noise-Gradient Perturbation of the Identity: the Formula, the Second-Order Expansion and the First Variation. Let $h\in L^{2}(\mu;X^{a})$ be the class of its noise gradient $\nabla_{a}\psi$, which is square-integrable with respect to $\mu$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient; it is the element written $\nabla_{a}(g\circ p_{n})$ in the hypothesis of A Bound on the Noise Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Weighted Fisher Information. By Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded the finitely many functions $\partial_{k}g$ and $\partial_{j}\partial_{i}g$ ($i,j,k\in[n]$) are bounded, so there are nonnegative $B,b\in\mathbb{R}$ with $|\partial_{k}g(u)|\le B$ and $a_{i}^{1/2}a_{j}^{1/2}|\partial_{j}\partial_{i}g(u)|\le b$ for all $u\in\mathbb{R}^{n}$ and $i,j,k\in[n]$. Let $\theta_{n}$ and $K_{n}$ be the constants of Relative Entropy along a Noise-Gradient Perturbation of the Identity: the Formula, the Second-Order Expansion and the First Variation and put

$$C_{g}=K_{n}b^{2}+B^{2}\sum_{k=1}^{n}\frac{a_{k}^{2}}{2c_{k}}\ge0,\qquad L=L^{a}_{\mu}(g),\qquad G=\langle\nabla\chi(\mu),h\rangle_{\mu}$$

(here $0\le K_{n}$ and $0<\theta_{n}$ by Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$ §expansion, read with $d=n$, where they are written $K_{d}$ and $c_{d}$). For $t\in\mathbb{R}$ let $S_{t}:X\to X$, $S_{t}(x)=x+t\,\nabla_{a}\psi(x)$, which is the map $\mathrm{id}+t\nabla_{a}\psi$ of that lemma, and put $\mu_{t}=(S_{t})_{\#}\mu$ and $\pi_{t}=(\mathrm{id},S_{t})_{\#}\mu$. By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §displacement, applied with $h$ and its representative $\nabla_{a}\psi$, $\pi_{t}\in\Pi^{a}(\mu,\mu_{t})$, $I^{a}(\pi_{t})=t^{2}\lVert h\rVert_{\mu}^{2}$, and $\mathcal{J}^{a}(\nabla\chi(\mu),\pi_{t})=t\,G$.

Step 2 (A bound on $G$). With $t=1$, the bound Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §bound gives $|G|=|\mathcal{J}^{a}(\nabla\chi(\mu),\pi_{1})|\le\lVert\nabla\chi(\mu)\rVert_{\mu}\sqrt{I^{a}(\pi_{1})}=\lVert\nabla\chi(\mu)\rVert_{\mu}\lVert h\rVert_{\mu}$.

Step 3 ($\lambda\beta L=G$). Let $\varepsilon\in\mathbb{R}$ be positive. First choose, by Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §differentiable applied to $\nabla\chi(\mu)$ and $\varepsilon$, a positive $\theta_{\chi}\in\mathbb{R}$ such that $|\chi(\nu)-\chi(\mu)-\mathcal{J}^{a}(\nabla\chi(\mu),\pi)|\le\varepsilon\sqrt{I^{a}(\pi)}$ for every $\nu\in\mathcal{P}^{a}_{\rho}$ and every $\pi\in\Pi^{a}(\mu,\nu)$ with $I^{a}(\pi)<\theta_{\chi}^{2}$. Then choose a positive $s\in\mathbb{R}$ with

$$2ns\,b\le1,\qquad s\,b\le\theta_{n},\qquad s\lVert h\rVert_{\mu}<\theta_{\chi},\qquad s\lVert h\rVert_{\mu}<r,\qquad\lambda\beta C_{g}\,s\le\varepsilon ,$$

which is possible since $\theta_{n}>0$ and each condition holds for all sufficiently small positive $s$. Let $t\in\{s,-s\}$, so $|t|=s$.

Since $2n|t|b\le1$, Relative Entropy along a Noise-Gradient Perturbation of the Identity: the Formula, the Second-Order Expansion and the First Variation §pushforward shows that $\mu_{t}$ has finite relative entropy with respect to $\gamma_{c}$, so $\mu_{t}\in\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain. As $\mu,\mu_{t}\in\mathcal{P}^{a}_{\rho}$ and $\pi_{t}\in\Pi^{a}(\mu,\mu_{t})$, The Noise Wasserstein Distance §distance (the pair $(\mu,\mu_{t})$ being noise-connected by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected) gives $W_{a}(\mu,\mu_{t})^{2}\le I^{a}(\pi_{t})=s^{2}\lVert h\rVert_{\mu}^{2}<r^{2}$, so $W_{a}(\mu,\mu_{t})<r$, both numbers being nonnegative. The local maximum therefore gives

$$\chi(\mu_{t})-\chi(\mu)\le\lambda\beta\bigl(H(\mu_{t}\,|\,\gamma_{c})-H(\mu\,|\,\gamma_{c})\bigr).$$

Since $I^{a}(\pi_{t})=s^{2}\lVert h\rVert_{\mu}^{2}<\theta_{\chi}^{2}$ and $\sqrt{I^{a}(\pi_{t})}=s\lVert h\rVert_{\mu}$, the choice of $\theta_{\chi}$ gives $|\chi(\mu_{t})-\chi(\mu)-tG|\le\varepsilon s\lVert h\rVert_{\mu}$, so $tG-\varepsilon s\lVert h\rVert_{\mu}\le\chi(\mu_{t})-\chi(\mu)$. Since $2n|t|b\le1$ and $|t|b\le\theta_{n}$, Relative Entropy along a Noise-Gradient Perturbation of the Identity: the Formula, the Second-Order Expansion and the First Variation §expansion gives $H(\mu_{t}\,|\,\gamma_{c})-H(\mu\,|\,\gamma_{c})\le tL+C_{g}s^{2}$. Combining the three inequalities, using $\lambda\beta>0$ and $\lambda\beta C_{g}s\le\varepsilon$,

$$tG-\varepsilon s\lVert h\rVert_{\mu}\le\lambda\beta\,tL+\lambda\beta C_{g}s^{2}\le\lambda\beta\,tL+\varepsilon s, \qquad\text{that is}\qquad t\,(G-\lambda\beta L)\le\varepsilon s\bigl(\lVert h\rVert_{\mu}+1\bigr).$$

Taking $t=s$ and $t=-s$ and dividing by $s>0$ gives $|G-\lambda\beta L|\le\varepsilon(\lVert h\rVert_{\mu}+1)$. As $\varepsilon>0$ was arbitrary, $G=\lambda\beta L$.

Step 4 (Conclusion). By Steps 3 and 2, with $R=(\lambda\beta)^{-1}\lVert\nabla\chi(\mu)\rVert_{\mu}$, a nonnegative real number not depending on $n$ or $g$,

$$\bigl|L^{a}_{\mu}(g)\bigr|=(\lambda\beta)^{-1}|G|\le R\,\lVert\nabla_{a}(g\circ p_{n})\rVert_{\mu}.$$

This holds for every $n\in\mathbb{N}$ and every $g\in C^{2}_{b}(\mathbb{R}^{n})$, and $\mu\in\mathcal{P}_{2}(X)$, so A Bound on the Noise Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Weighted Fisher Information §fisher shows that $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$. Hence $\mu\in\mathcal{D}_{\Sigma}$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain, and the pair has regular penalised maxima.