Proof of The Gaussian Log-Sobolev, HWI and Poincare Inequalities in the Noise Geometry, in Entropy Form and in Gross's Functional Form

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

By Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §background the setting A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation is in force, with reference measure $\rho=\gamma_{c}$; it is layered on Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, as is First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, so the results stated in the latter setting apply with $\rho=\gamma_{c}$. Throughout, $\phi$ 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 and $\kappa$ is as in the statement.

Step 1 (The Gaussian entropy pair). Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $\beta=1$; its hypothesis holds with the given $\kappa$. By The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain, $\mathcal{D}$ is the set of the $\mu\in\mathcal{P}(X)$ of finite relative entropy with respect to $\gamma_{c}$; by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain, $\mathcal{D}_{\Sigma}$ is the set of the $\mu\in\mathcal{D}$ that have a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$; and by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair, $\mathcal{E}(\mu)=H(\mu\,|\,\gamma_{c})$ for $\mu\in\mathcal{D}$ and $\Sigma(\mu)=Z^{a}_{\mu}$ for $\mu\in\mathcal{D}_{\Sigma}$, the noise score field of $\mu$. By The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field, $\lVert\Sigma(\mu)\rVert_{\mu}^{2}=\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ for $\mu\in\mathcal{D}_{\Sigma}$. The quadruple 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, and it is $\lambda$-displacement convex with $\lambda=1/\kappa$, a positive real number, by The Gaussian Entropy Pair is Uniformly Displacement Convex, with Modulus the Temperature over the Variance-to-Noise Bound §convex with $\beta=1$.

Step 2 (The reference measure is a point of zero score). By The Integral of an Indicator Function is the Measure of the Set, $\int_{X}\mathbf{1}_{A}\cdot1\,d\gamma_{c}=\gamma_{c}(A)$ for $A\in\mathcal{B}(X)$, so the constant function $1$ is a density of $\gamma_{c}$ with respect to $\gamma_{c}$ in the sense of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities. Since $\exp(0)=1$ by claim 1 of Basic Properties of the Exponential Function, $\log1=\log(\exp(0))=0$ by The Natural Logarithm, so $\phi\circ1$ is the constant $\phi(1)=1\cdot\log1=0$, which is integrable with integral $0$: by The Integral of an Indicator Function is the Measure of the Set with $A=X$, the constant $1=\mathbf{1}_{X}$ has $\int_{X}1\,d\gamma_{c}=\gamma_{c}(X)=1$, so it is integrable, and its multiple by $0$ is integrable with integral $0$ by Linearity and Monotonicity of the Lebesgue Integral §integrable. By Relative Entropy of Probability Measures §relative-entropy, $\gamma_{c}$ has finite relative entropy with respect to $\gamma_{c}$ and $H(\gamma_{c}\,|\,\gamma_{c})=0$; thus $\gamma_{c}\in\mathcal{D}$ and $\mathcal{E}(\gamma_{c})=0$. By The Relative Score on a Hilbert Space: the Gaussian Measure Has Score Zero, and the Score as a Square-Integrable Field in the Weighted Sequence Space §gaussian, $\gamma_{c}\in\mathcal{P}_{2}(X)$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, with $\mathcal{I}_{a}(\gamma_{c}\,|\,\gamma_{c})=0$. Hence $\gamma_{c}\in\mathcal{D}_{\Sigma}$ and, by Step 1, $\lVert\Sigma(\gamma_{c})\rVert_{\gamma_{c}}^{2}=0$; as the norm of the real Hilbert space $L^{2}(\gamma_{c};X^{a})$ of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields vanishes only at the zero vector, $\Sigma(\gamma_{c})$ is the zero vector of $L^{2}(\gamma_{c};X^{a})$.

Step 3 (Claim 1). Let $\mu$ be as in claim 1. Then $\mu\in\mathcal{D}$, $\mu\in\mathcal{P}_{2}(X)$ by Relative Entropy with Respect to a Diagonal Gaussian Measure on a Hilbert Space: the Moment Bound, the Cutoff Projections, and Bounded, Tight, Weakly Closed, Wasserstein-Closed and Weakly Sequentially Compact Sublevel Sets §moment, and $\mu$ has a relative score and finite Fisher information, so $\mu\in\mathcal{D}_{\Sigma}$. By Steps 1 and 2, Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score §log-sobolev applies to the pair of Step 1 with $\lambda=1/\kappa$ and $\mu_{*}=\gamma_{c}$, and gives

$$H(\mu\,|\,\gamma_{c})-0\le\frac{\kappa}{2}\,\lVert\Sigma(\mu)\rVert_{\mu}^{2}=\frac{\kappa}{2}\,\mathcal{I}_{a}(\mu\,|\,\gamma_{c}).$$

Step 4 (Claim 2). Let $\mu$ be as in claim 1. Then $\mu\in\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property §inclusion, and $\gamma_{c}=\rho\in\mathcal{P}^{a}_{\rho}$ 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 §reference. As in Step 3, $\mu\in\mathcal{D}_{\Sigma}$, and Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score §hwi with $\lambda=1/\kappa$ and $\mu_{*}=\gamma_{c}$ gives $H(\mu\,|\,\gamma_{c})-0\le\lVert\Sigma(\mu)\rVert_{\mu}W_{a}(\mu,\gamma_{c})-\frac{1}{2\kappa}W_{a}(\mu,\gamma_{c})^{2}$. Since $\lVert\Sigma(\mu)\rVert_{\mu}$ is nonnegative with square $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ by Step 1, it equals $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})^{1/2}$ by Existence and Uniqueness of the Nonnegative Square Root, which proves claim 2.

Step 5 (Quadratic expressions in a cylindrical function). Let $F\in\mathcal{F}C^{1}_{b}(X)$ have a representation $(n,\psi)$, and let $\alpha,s,b_{0}\in\mathbb{R}$. We show that $\alpha+sF+b_{0} F^{2}\in\mathcal{F}C^{1}_{b}(X)$ and that for every $k\in\mathbb{N}$

$$\partial_{k}\bigl(\alpha+sF+b_{0} F^{2}\bigr)=(s+2b_{0} F)\,\partial_{k}F\quad\text{on }X,\qquad\text{and both sides vanish for }k>n .$$

By Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, $\psi$ is of class $C^{1}$ on $\mathbb{R}^{n}$ and there are real $B\ge0$ and $B_{i}\ge0$ with $|\psi|\le B$ and $|\partial_{i}\psi|\le B_{i}$ for $i\in[n]$; by clause 1 of C^k Maps on a Euclidean Open Set, read through its clause 3 as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, $\psi$ and each $\partial_{i}\psi$ are continuous at every point of $\mathbb{R}^{n}$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces, which for real-valued maps reads: for every $\varepsilon>0$ there is $\delta>0$ with $|f(y)-f(x)|<\varepsilon$ whenever $\lVert y-x\rVert<\delta$. Let $\chi=\alpha+s\psi+b_{0}\psi^{2}:\mathbb{R}^{n}\to\mathbb{R}$. It is bounded by $|\alpha|+|s|B+|b_{0}|B^{2}$. It is continuous at every $x$: since $\psi(y)^{2}-\psi(x)^{2}=(\psi(y)+\psi(x))(\psi(y)-\psi(x))$, we have $|\chi(y)-\chi(x)|\le(|s|+2|b_{0}|B)\,|\psi(y)-\psi(x)|$, and it suffices to take $\delta$ for $\psi$ at $x$ with $\varepsilon/(|s|+2|b_{0}|B+1)$ in place of $\varepsilon$. Fix $x\in\mathbb{R}^{n}$ and $i\in[n]$, let $e_{i}$ be the $i$-th standard basis vector of $\mathbb{R}^{n}$, and let $u:\mathbb{R}\to\mathbb{R}$, $u(t)=\psi(x+te_{i})$. The real line is an interval of which $0$ is an interior point by claim 1 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line, and the condition of Partial Derivative on a Euclidean Open Set for $\psi$ at $x$ in the $i$-th variable is, word for word, the condition of Derivative at an Interior Point for $u$ at $0$, the extra clauses of the two definitions being vacuous here: the requirement $x+he_{i}\in\mathbb{R}^{n}$ of the former holds for every $h$, the open set being all of $\mathbb{R}^{n}$, and the restriction $0+h\in\mathbb{R}$ of the latter excludes no $h$, the interval being all of $\mathbb{R}$; so $u$ is differentiable at $0$ with $u'(0)=\partial_{i}\psi(x)$. By claims 1 to 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $v=\alpha+su+b_{0} uu$ is differentiable at $0$ with $v'(0)=su'(0)+2b_{0} u(0)u'(0)$, and since $v(t)=\chi(x+te_{i})$, reading the same identification backwards gives that $\partial_{i}\chi(x)$ exists and equals $(s+2b_{0}\psi(x))\,\partial_{i}\psi(x)$. The function $\partial_{i}\chi=(s+2b_{0}\psi)\partial_{i}\psi$ is bounded by $(|s|+2|b_{0}|B)B_{i}$, and is continuous at every $x$ because

$$|\partial_{i}\chi(y)-\partial_{i}\chi(x)|\le(|s|+2|b_{0}|B)\,|\partial_{i}\psi(y)-\partial_{i}\psi(x)|+2|b_{0}|B_{i}\,|\psi(y)-\psi(x)|,$$

so it suffices to take the smaller of the two $\delta$'s for $\partial_{i}\psi$ and $\psi$ at $x$ with $\varepsilon/(2(|s|+2|b_{0}|B)+1)$ and $\varepsilon/(4|b_{0}|B_{i}+1)$ in place of $\varepsilon$. Hence $\chi$ is of class $C^{1}$ on $\mathbb{R}^{n}$ by clause 1 of C^k Maps on a Euclidean Open Set, and $\chi\in C^{1}_{b}(\mathbb{R}^{n})$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded. Since $\chi\circ p_{n}=\alpha+sF+b_{0} F^{2}$, this function lies in $\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, with representation $(n,\chi)$, and Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial, applied to both representations $(n,\chi)$ and $(n,\psi)$, gives for $k\le n$ that $\partial_{k}(\chi\circ p_{n})=(\partial_{k}\chi)\circ p_{n}=(s+2b_{0} F)\,(\partial_{k}\psi)\circ p_{n}=(s+2b_{0} F)\,\partial_{k}F$, and for $k>n$ that both $\partial_{k}(\chi\circ p_{n})$ and $\partial_{k}F$ vanish identically. In particular, by the display of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient computed with these representations, $|\nabla_{a}(\alpha+sF+b_{0} F^{2})|_{a}^{2}=\sum_{k=1}^{n}a_{k}(s+2b_{0} F)^{2}(\partial_{k}F)^{2}$.

Step 6 (Entropies of bounded functions). Let $G:X\to\mathbb{R}$ be Borel with values in an interval $[0,b]$. The function $\phi$ is continuous on $[0,\infty)$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous, hence on $[0,b]$, so Extreme Value Theorem on a Closed Real Interval, applied to the restriction of $\phi$ to $[0,b]$, gives $r_{\min},r_{\max}\in[0,b]$ with $\phi(r_{\min})\le\phi(r)\le\phi(r_{\max})$ for $r\in[0,b]$; with $M_{b}=\max(|\phi(r_{\min})|,|\phi(r_{\max})|)$ we get $|\phi(r)|\le M_{b}$ for $r\in[0,b]$; $\phi\circ G$ is 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. Thus $G$ and $\phi\circ G$ are bounded and Borel, hence integrable with respect to every Borel probability measure on $X$ by Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §functional, and $\operatorname{Ent}_{\gamma_{c}}(G)$ is defined by The Entropy of a Nonnegative Function with Respect to a Probability Measure §entropy. For a real constant $r$, $\int_{X}r\,d\gamma_{c}=r$ by The Integral of an Indicator Function is the Measure of the Set and Linearity and Monotonicity of the Lebesgue Integral §integrable, $\gamma_{c}$ being a probability measure. Now let $F\in\mathcal{F}C^{1}_{b}(X)$ with representation $(n,\psi)$, and fix a real $B\ge1$ with $|\psi|\le B$, so that $|F|\le B$ on $X$. Since $F$ is Borel by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel, $F^{2}$ is Borel by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, nonnegative and bounded by $B^{2}$; so $F^{2}$ and $\phi\circ F^{2}$ are integrable with respect to $\gamma_{c}$, which is the first part of claim 3. We write $D=\int_{X}|\nabla_{a}F|_{a}^{2}\,d\gamma_{c}$, a nonnegative real number by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient.

Step 7 (The inequality for $F^{2}+\varepsilon$). Let $F$, $(n,\psi)$, $B$ and $D$ be as in Step 6, and let $\varepsilon$ be real with $0<\varepsilon\le1$. Let $g=\varepsilon+F^{2}$. By Step 5 (with $\alpha=\varepsilon$, $s=0$, $b_{0}=1$), $g\in\mathcal{F}C^{1}_{b}(X)$ and $\partial_{k}g=2F\,\partial_{k}F$ for every $k$, with $\partial_{k}g=0$ for $k>n$; and $\varepsilon\le g\le B^{2}+1$. Let $N_{\varepsilon}=\int_{X}g\,d\gamma_{c}=\varepsilon+\int_{X}F^{2}\,d\gamma_{c}$, so $\varepsilon\le N_{\varepsilon}\le B^{2}+1$. The function $g/N_{\varepsilon}:X\to[0,\infty)$ is Borel, so by claim 3 of Image Measures, Measures with Densities, and Change of Variables the measure $\mu$ with density $g/N_{\varepsilon}$ with respect to $\gamma_{c}$, $\mu(A)=\int_{X}\mathbf{1}_{A}\,(g/N_{\varepsilon})\,d\gamma_{c}$, is a measure on $\mathcal{B}(X)$, with $\mu(X)=N_{\varepsilon}^{-1}\int_{X}g\,d\gamma_{c}=1$; so $\mu\in\mathcal{P}(X)$ and $g/N_{\varepsilon}$ is a density of $\mu$ with respect to $\gamma_{c}$ in the sense of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities. We use repeatedly that, by claim 3 of Image Measures, Measures with Densities, and Change of Variables, a Borel $f:X\to\mathbb{R}$ with $f\,g/N_{\varepsilon}$ integrable with respect to $\gamma_{c}$ is integrable with respect to $\mu$, with $\int_{X}f\,d\mu=N_{\varepsilon}^{-1}\int_{X}f\,g\,d\gamma_{c}$.

Entropy. The function $g/N_{\varepsilon}$ is Borel with values in $[0,(B^{2}+1)/\varepsilon]$, so $\phi\circ(g/N_{\varepsilon})$ is integrable with respect to $\gamma_{c}$ by Step 6, and $\mu$ has finite relative entropy with respect to $\gamma_{c}$, with $H(\mu\,|\,\gamma_{c})=\int_{X}\phi\circ(g/N_{\varepsilon})\,d\gamma_{c}$, by Relative Entropy of Probability Measures §relative-entropy. For $r>0$, $\log r=\log(r/N_{\varepsilon})+\log N_{\varepsilon}$ by The Natural Logarithm, so $\phi(r/N_{\varepsilon})=(r/N_{\varepsilon})\log(r/N_{\varepsilon})=N_{\varepsilon}^{-1}\bigl(\phi(r)-r\log N_{\varepsilon}\bigr)$. Applying this with $r=g(x)>0$ and integrating, using Linearity and Monotonicity of the Lebesgue Integral §integrable, Step 6 for $g$, and The Entropy of a Nonnegative Function with Respect to a Probability Measure §entropy,

$$H(\mu\,|\,\gamma_{c})=N_{\varepsilon}^{-1}\Bigl(\int_{X}\phi\circ g\,d\gamma_{c}-N_{\varepsilon}\log N_{\varepsilon}\Bigr)=N_{\varepsilon}^{-1}\Bigl(\int_{X}\phi\circ g\,d\gamma_{c}-\phi(N_{\varepsilon})\Bigr)=N_{\varepsilon}^{-1}\operatorname{Ent}_{\gamma_{c}}(g).$$

In particular $\mu\in\mathcal{P}_{2}(X)$ by Relative Entropy with Respect to a Diagonal Gaussian Measure on a Hilbert Space: the Moment Bound, the Cutoff Projections, and Bounded, Tight, Weakly Closed, Wasserstein-Closed and Weakly Sequentially Compact Sublevel Sets §moment.

Relative score. For $k\in\mathbb{N}$ let $\zeta_{k}=\partial_{k}g\cdot(1/g)$. The function $1/g$ is Borel by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable: $\{1/g>r\}$ is $X$ for $r\le0$ and $\{-g>-1/r\}$ for $r>0$, Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. As $\partial_{k}g$ is bounded and Borel by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel and $0<1/g\le1/\varepsilon$, $\zeta_{k}$ is bounded and Borel (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); so $\zeta_{k}^{2}$ is integrable with respect to $\mu$ by Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §functional, and $\zeta_{k}$ defines an element of $L^{2}(\mu)$, again written $\zeta_{k}$; for $k>n$, $\zeta_{k}=0$. Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and put $q(x)=x_{k}\varphi(x)/c_{k}-\partial_{k}\varphi(x)$. By The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, then the density formula (the function $\zeta_{k}\varphi\,g/N_{\varepsilon}=N_{\varepsilon}^{-1}\partial_{k}g\,\varphi$ being bounded and Borel), then Gaussian Integration by Parts for Products of Cylindrical Functions, and Closability of the Noise Gradient in the Gaussian Lebesgue Space §ibp applied with $g$ in place of $F$ and with $\varphi$, then Linearity and Monotonicity of the Lebesgue Integral §integrable and the density formula once more (the function $q\,g/N_{\varepsilon}$ being integrable with respect to $\gamma_{c}$ as $N_{\varepsilon}^{-1}$ times the integrable function $g\,q$ of Gaussian Integration by Parts for Products of Cylindrical Functions, and Closability of the Noise Gradient in the Gaussian Lebesgue Space §ibp),

$$\langle\zeta_{k},\varphi\rangle_{L^{2}(\mu)}=\int_{X}\zeta_{k}\varphi\,d\mu=\frac{1}{N_{\varepsilon}}\int_{X}\partial_{k}g\,\varphi\,d\gamma_{c}=\frac{1}{N_{\varepsilon}}\int_{X}g\,q\,d\gamma_{c}=\int_{X}\Bigl(\frac{x_{k}}{c_{k}}\,\varphi(x)-\partial_{k}\varphi(x)\Bigr)\mu(dx).$$

By The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score, $\mu$ has a relative score with respect to $\gamma_{c}$, with components $\zeta_{k}$.

Fisher information. For $k>n$ the term $a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$ is $0$, so the partial sums of $\sum_{k=1}^{\infty}a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$ are constant from the $n$-th on; the series converges with sum $\sum_{k=1}^{n}a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$, and by Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §information $\mu$ has finite Fisher information relative to $\gamma_{c}$ with weights $a$ and $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ is that finite sum. For $k\le n$, by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and the density formula, $\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}=\int_{X}\zeta_{k}^{2}\,d\mu=N_{\varepsilon}^{-1}\int_{X}(\partial_{k}g)^{2}/g\,d\gamma_{c}$, and pointwise

$$\frac{(\partial_{k}g)^{2}}{g}=\frac{4F^{2}(\partial_{k}F)^{2}}{\varepsilon+F^{2}}\le4(\partial_{k}F)^{2},$$

since $0\le F^{2}\le\varepsilon+F^{2}$. By the monotonicity and linearity of Linearity and Monotonicity of the Lebesgue Integral §integrable and the display of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient for the representation $(n,\psi)$,

$$\mathcal{I}_{a}(\mu\,|\,\gamma_{c})\le\frac{4}{N_{\varepsilon}}\sum_{k=1}^{n}a_{k}\int_{X}(\partial_{k}F)^{2}\,d\gamma_{c}=\frac{4}{N_{\varepsilon}}\int_{X}|\nabla_{a}F|_{a}^{2}\,d\gamma_{c}=\frac{4D}{N_{\varepsilon}}.$$

Conclusion. By claim 1 (Step 3) applied to $\mu$, $N_{\varepsilon}^{-1}\operatorname{Ent}_{\gamma_{c}}(g)=H(\mu\,|\,\gamma_{c})\le\frac{\kappa}{2}\cdot\frac{4D}{N_{\varepsilon}}$, and multiplying by $N_{\varepsilon}>0$,

$$\operatorname{Ent}_{\gamma_{c}}(F^{2}+\varepsilon)\le2\kappa D\qquad(0<\varepsilon\le1).$$

Step 8 (Claim 3: letting $\varepsilon\to0$). Let $F$, $B$, $D$ be as in Step 6, and for $j\in\mathbb{N}$ let $g_{j}=F^{2}+1/j$, with values in $[0,B^{2}+1]$. The sequence $(1/j)_{j}$ converges to $0$ in the sense of Limit of a Sequence of Real Numbers: given a real $\theta>0$, claim 3 of The Archimedean Property of the Real Numbers gives $j_{0}\in\mathbb{N}$ with $0<1/j_{0}<\theta$, and then $0<1/j\le1/j_{0}<\theta$ for $j\ge j_{0}$. As a constant sequence converges to its value, Arithmetic of Limits of Real Sequences §sums shows that for each $x\in X$ the sequence $(F(x)^{2}+1/j)_{j}$ in $[0,\infty)$ converges to $F(x)^{2}$, so $(\phi(g_{j}(x)))_{j}$ converges to $\phi(F(x)^{2})$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential and the continuity of $\phi$ (The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous); moreover $|\phi\circ g_{j}|\le M_{B^{2}+1}$ (Step 6), a constant, which is integrable. By Dominated Convergence Theorem, $\int_{X}\phi\circ g_{j}\,d\gamma_{c}\to\int_{X}\phi\circ F^{2}\,d\gamma_{c}$. Also $\int_{X}g_{j}\,d\gamma_{c}=\int_{X}F^{2}\,d\gamma_{c}+1/j\to\int_{X}F^{2}\,d\gamma_{c}$ by the same sum clause, so $\phi\bigl(\int_{X}g_{j}\,d\gamma_{c}\bigr)\to\phi\bigl(\int_{X}F^{2}\,d\gamma_{c}\bigr)$ by the same continuity. By Arithmetic of Limits of Real Sequences §scalar, $\operatorname{Ent}_{\gamma_{c}}(g_{j})\to\operatorname{Ent}_{\gamma_{c}}(F^{2})$; since $\operatorname{Ent}_{\gamma_{c}}(g_{j})\le2\kappa D$ for every $j$ by Step 7, claim 1 of Order Properties of Limits of Real Sequences gives $\operatorname{Ent}_{\gamma_{c}}(F^{2})\le2\kappa D$, which completes claim 3.

Step 9 (A second-order expansion of $\phi$ at $1$). Let $U=(0,\infty)$, open in $\mathbb{R}^{1}$ by Open Subset of Euclidean Space (for $t\in U$, every $y$ with $(y-t)^{2}<t^{2}$ is positive), and let $f:U\to\mathbb{R}$, $f(t)=t\log t=\phi(t)$. Every $t\in U$ is an interior point of the interval $U$, as $t/2<t<2t$. We first show that $\log$, defined in The Natural Logarithm, is differentiable at every $t\in U$ in the sense of Derivative at an Interior Point, with derivative $1/t$. Fix $t\in U$ and a real $h$ with $0<|h|<t/2$, and put $\tau=1+h/t$, so that $\tau>1/2$ and $t+h=t\tau$. By the identity $\log(t\tau)=\log t+\log\tau$ recorded in The Natural Logarithm, $\log(t+h)-\log t=\log\tau$, and by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log, $1-\tau^{-1}\le\log\tau\le\tau-1$, that is, $h/(t+h)\le\log(t+h)-\log t\le h/t$. Dividing by $h$, the difference quotient $Q_{h}=(\log(t+h)-\log t)/h$ lies between $1/(t+h)$ and $1/t$ (in this order if $h>0$, in the reverse order if $h<0$), so, using $t+h>t/2$,

$$\Bigl|Q_{h}-\frac{1}{t}\Bigr|\le\Bigl|\frac{1}{t+h}-\frac{1}{t}\Bigr|=\frac{|h|}{t\,(t+h)}\le\frac{2|h|}{t^{2}}.$$

Given a real $\varepsilon>0$, put $\delta_{t}=\min(t/2,\varepsilon t^{2}/2)$; every $h$ with $0<|h|<\delta_{t}$ then satisfies $t+h\in U$ and $|Q_{h}-1/t|<2\delta_{t}/t^{2}\le\varepsilon$, which is the condition of Derivative at an Interior Point with $L=1/t$. The identity map of $U$ has derivative $1$ (its difference quotients equal $1$); so by claims 1 to 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $f$ is differentiable at every $t\in U$ with $f'(t)=\log t+1$, and $f'$ is differentiable at every $t$ with derivative $\omega(t)=1/t$, which is differentiable at every $t\in U$ by claim 2 of Reciprocal Rule for One-Dimensional Derivatives, $0$ not lying in $U$. Two elementary observations transfer this to the notions used by C^k Maps on a Euclidean Open Set. First, if $\hat\kappa:U\to\mathbb{R}$ is differentiable at $t$ with derivative $L$ in the sense of Derivative at an Interior Point, then the partial derivative of $\hat\kappa$ with respect to the first variable exists at $t$ with value $L$ in the sense of Partial Derivative on a Euclidean Open Set: the two conditions concern the same difference quotient, and replacing $\delta$ by $\min(\delta,t)$ ensures $t+h\in U$ whenever $0<|h|<\delta$. Second, such a $\hat\kappa$ is continuous at $t$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces: taking $\delta_{1}\le t$ for the value $1$ in the derivative condition, $|\hat\kappa(t+h)-\hat\kappa(t)|\le|h|(|L|+1)$ for $0<|h|<\delta_{1}$, which is less than a given $\varepsilon>0$ when also $|h|<\varepsilon/(|L|+1)$. Hence $f$, $\partial_{1}f=f'$ and $\partial_{1}\partial_{1}f=\omega$ exist and are continuous at every point of $U$; by clause 1 of C^k Maps on a Euclidean Open Set, $f$ and $\partial_{1}f$ are of class $C^{1}$ on $U$, and by its clause 2 (with $k=1$), $f$ is of class $C^{2}$ on $U$. Moreover $f(1)=0$, $\partial_{1}f(1)=\log1+1=1$ and $\partial_{1}\partial_{1}f(1)=1$, using $\log1=0$ from Step 2. By Second-Order Taylor Expansion with Peano Remainder with $n=1$ and $x=1$, where the Euclidean norm of $h\in\mathbb{R}^{1}$ is $|h|$: for every real $\eta>0$ there is a real $\delta>0$ such that every real $h$ with $|h|<\delta$ satisfies $1+h>0$ and

$$\Bigl|\phi(1+h)-h-\tfrac{1}{2}h^{2}\Bigr|\le\eta\,h^{2}.$$

For $h>-1$ write $R(h)=\phi(1+h)-h-\frac{1}{2}h^{2}$.

Step 10 (Claim 4). Let $F$, $(n,\psi)$, $B\ge1$ and $D$ be as in Step 6, and put $m=\int_{X}F\,d\gamma_{c}$, $v=\int_{X}F^{2}\,d\gamma_{c}$, $m_{3}=\int_{X}F^{3}\,d\gamma_{c}$ and $m_{4}=\int_{X}F^{4}\,d\gamma_{c}$, integrals of bounded Borel functions (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); by Linearity and Monotonicity of the Lebesgue Integral §integrable, $|m|\le B$, $0\le v\le B^{2}$, $|m_{3}|\le B^{3}$ and $m_{4}\ge0$. Let $\eta>0$, let $\delta$ be as in Step 9, and let $s=\min(\eta,1/B,\delta/(6B))$, so that $0<s\le\eta$, $sB\le1$ and $s\le1$. Let $G_{s}=1+sF$. By Step 5 (with $\alpha=1$, $s$, $b_{0}=0$), $G_{s}\in\mathcal{F}C^{1}_{b}(X)$ with $|\nabla_{a}G_{s}|_{a}^{2}=s^{2}\sum_{k=1}^{n}a_{k}(\partial_{k}F)^{2}=s^{2}|\nabla_{a}F|_{a}^{2}$, so claim 3 applied to $G_{s}$ gives, with Linearity and Monotonicity of the Lebesgue Integral §integrable,

$$\operatorname{Ent}_{\gamma_{c}}(G_{s}^{2})\le2\kappa s^{2}D .$$

Now $G_{s}^{2}=1+u$ with $u=2sF+s^{2}F^{2}$, and $|u|\le2sB+s^{2}B^{2}\le3sB<\delta$ on $X$. Let $w=\int_{X}u\,d\gamma_{c}=2sm+s^{2}v$; then $|w|\le\int_{X}|u|\,d\gamma_{c}\le3sB<\delta$ and $\int_{X}G_{s}^{2}\,d\gamma_{c}=1+w$. By Step 9, $\phi\circ G_{s}^{2}=u+\frac{1}{2}u^{2}+R\circ u$ on $X$, where $R\circ u$ is integrable as a difference of integrable functions (Step 6) and satisfies $R\circ u\ge-\eta u^{2}$; and $\phi(1+w)=w+\frac{1}{2}w^{2}+R(w)$ with $R(w)\le\eta w^{2}$. By The Entropy of a Nonnegative Function with Respect to a Probability Measure §entropy and Linearity and Monotonicity of the Lebesgue Integral §integrable,

$$\operatorname{Ent}_{\gamma_{c}}(G_{s}^{2})=\frac{1}{2}\Bigl(\int_{X}u^{2}\,d\gamma_{c}-w^{2}\Bigr)+\int_{X}R\circ u\,d\gamma_{c}-R(w)\ge\frac{1}{2}\Bigl(\int_{X}u^{2}\,d\gamma_{c}-w^{2}\Bigr)-\eta\Bigl(\int_{X}u^{2}\,d\gamma_{c}+w^{2}\Bigr).$$

Here $\int_{X}u^{2}\,d\gamma_{c}\le9s^{2}B^{2}$ and $w^{2}\le9s^{2}B^{2}$, while expanding the squares,

$$\frac{1}{2}\Bigl(\int_{X}u^{2}\,d\gamma_{c}-w^{2}\Bigr)=2s^{2}(v-m^{2})+2s^{3}(m_{3}-mv)+\frac{s^{4}}{2}(m_{4}-v^{2})\ge2s^{2}(v-m^{2})-4s^{3}B^{3}-\frac{s^{4}}{2}B^{4}.$$

Combining the last three displays and dividing by $2s^{2}>0$,

$$v-m^{2}\le\kappa D+2sB^{3}+\frac{s^{2}}{4}B^{4}+9\eta B^{2}\le\kappa D+\eta\Bigl(2B^{3}+\frac{B^{4}}{4}+9B^{2}\Bigr),$$

using $s^{2}\le s\le\eta$. Let $C_{0}=2B^{3}+\frac{B^{4}}{4}+9B^{2}$ be the bracket, which does not depend on $\eta$ and is positive as $B\ge1$. Given a positive real $\varepsilon$, the argument above with $\eta=\varepsilon/C_{0}$ gives $v-m^{2}\le\kappa D+\varepsilon$; as $\varepsilon$ was arbitrary, Comparison of Real Numbers with Arbitrary Positive Slack §slack-above gives $v-m^{2}\le\kappa D$, which is claim 4.