Proof of Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery)

Each result cited is universally quantified over the data in its own statement. Elementary real arithmetic and order (The Real Numbers: Standing Notation and Background §background) are used without citation; this covers manipulations of inequalities, finite sums and products, the limit laws for convergent real sequences, the preservation of non-strict inequalities under limits, and the fact that $r\le s$ when $r,s$ are nonnegative with $r^{2}\le s^{2}$. Integrals and integrability are those of Measure Spaces and the Lebesgue Integral: Standing Notation; linearity and monotonicity of the integral are Linearity and Monotonicity of the Lebesgue Integral §integrable and Linearity and Monotonicity of the Lebesgue Integral §nonnegative; a bounded Borel function is integrable with respect to a Borel probability measure on $X$ by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space. Test functions in $\mathcal{F}C^{1}_{b}(X)$ are written $\varphi$, $u$; the letter $\phi$ is reserved for 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.

Notation. Write $\gamma^{V}=\gamma^{V}_{\beta}$ and $w=w_{V,\beta}$ for the weight of The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §weight, and keep the full notation $Z_{V,\beta}$ for the normaliser of The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §normaliser; $Z_{V,\beta}$ is positive and $\gamma^{V}\in\mathcal{P}(X)$ by The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §gibbs. Let $V=v\circ p_{d}$ with head dimension $d$ and profile $v$ (Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §admissible), and fix constants $b$ and $C$ as in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §below and Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §slope; $C\ge0$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §translation, and $-b\le V(x)$ for every $x$, $V$ and every $\partial_{k}V$ being continuous, hence Borel, by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity. Let $\Theta:\mathbb{R}\to\mathbb{R}$, $\Theta(s)=\exp(-s/\beta)$, so that $w=\Theta\circ V$. Put

$$\lambda=\frac{\beta}{\kappa}-K=\frac{\beta-\kappa K}{\kappa},$$

which is positive since $\kappa K<\beta$ and $\kappa>0$.

Step 1 (Calculus of profiles). For $N\in\mathbb{N}$ the set $\mathbb{R}^{N}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and the class $C^{1}$ and the partial derivatives on it are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, that is, of C^k Maps on a Euclidean Open Set, which are the notions of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$. Points of $\mathbb{R}^{N}$ are written $y,y'$.

(a) Sums and products. Let $f,g:\mathbb{R}^{N}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{N}$ and $t\in\mathbb{R}$. By claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied to $g$ and the scalar $t$, then to $f$ and $tg$, and to $f$ and $g$, the functions $f+tg$ and $fg$ are of class $C^{1}$ on $\mathbb{R}^{N}$. The partial derivatives of $f$ and $g$ exist at every point by clause 1 of C^k Maps on a Euclidean Open Set, read through its clause 3, so by claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied in the same way, $\partial_{i}(f+tg)=\partial_{i}f+t\,\partial_{i}g$ and $\partial_{i}(fg)=\partial_{i}f\,g+f\,\partial_{i}g$ pointwise on $\mathbb{R}^{N}$, for every $i\in[N]$. By these formulas, $C^{1}_{b}(\mathbb{R}^{N})$ (Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded) is closed under sums, real multiples and products.

(b) Lifting. Let $n\le N$, $\pi:\mathbb{R}^{N}\to\mathbb{R}^{n}$, $\pi(y)=(y_{1},\dots,y_{n})$, and let $f:\mathbb{R}^{n}\to\mathbb{R}$ be of class $C^{1}$. For $y\in\mathbb{R}^{N}$ the slice of $f\circ\pi$ at $y$ in the $i$-th variable is the slice of $f$ at $\pi(y)$ if $i\le n$ (both taken on a common interval, every radius being admissible in claim 1 of Slice Function and the Partial Derivative, the domains being whole Euclidean spaces) and is constant if $n<i\le N$; so by claim 2 of Slice Function and the Partial Derivative and claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\partial_{i}(f\circ\pi)=(\partial_{i}f)\circ\pi$ for $i\le n$ and $\partial_{i}(f\circ\pi)=0$ for $n<i\le N$. The coordinate functions $y\mapsto y_{l}$ ($l\in[n]$) of $\pi$ are smooth, hence of class $C^{1}$, on $\mathbb{R}^{N}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and Smooth Map on a Euclidean Open Set; so $\pi$ is of class $C^{1}$ on $\mathbb{R}^{N}$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set, and $f\circ\pi$ is of class $C^{1}$ on $\mathbb{R}^{N}$ by claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, applied with $F=\pi$, the open set $\mathbb{R}^{n}$ in place of $V$, and $G=f$. By the formulas just obtained, $f\circ\pi$ lies in $C^{1}_{b}(\mathbb{R}^{N})$ if $f\in C^{1}_{b}(\mathbb{R}^{n})$. Since $p_{n}=\pi\circ p_{N}$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $f\circ p_{n}=(f\circ\pi)\circ p_{N}$; in particular every $\varphi\in\mathcal{F}C^{1}_{b}(X)$ with representation $(n,\psi)$ has the representation $(N,\psi\circ\pi)$ for every $N\ge n$.

(c) Composition with $\Theta$. Let $f:\mathbb{R}^{N}\to\mathbb{R}$ be of class $C^{1}$. By Derivative and Continuity of the Scaled Exponential Function with $c=-1/\beta$, $\Theta$ is differentiable at every point of the interval $\mathbb{R}$ with $\Theta'(s)=-\beta^{-1}\Theta(s)$, and continuous on $(\mathbb{R},d_{\mathbb{R}})$. The slice of $\Theta\circ f$ at $y$ in the $i$-th variable is $\Theta$ composed with the slice of $f$, so by Chain Rule for One-Dimensional Derivatives and claim 2 of Slice Function and the Partial Derivative, $\partial_{i}(\Theta\circ f)=-\beta^{-1}(\Theta\circ f)\,\partial_{i}f$. For a real-valued function on $\mathbb{R}^{q}$ ($q\in\mathbb{N}$), continuity at a point in the sense of Continuity at a Point for Maps Between Euclidean Spaces is the same as continuity at that point from $(\mathbb{R}^{q},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$ in the sense of Continuous Map Between Metric Spaces, since $d_{E}(y,y')$ is the nonnegative square root of $\sum_{i}(y_{i}-y'_{i})^{2}$, $|s|^{2}=s^{2}$, and $r<\delta$ exactly when $r^{2}<\delta^{2}$ for nonnegative $r$ and positive $\delta$. So $\Theta$ (with $q=1$) is continuous at every point in the first sense, and $\Theta\circ f$ is continuous at every point by clause 1 of C^k Maps on a Euclidean Open Set and Composition of Continuous Euclidean Maps; $\Theta\circ f$ and $\partial_{i}f$ (clause 1 of C^k Maps on a Euclidean Open Set) are then continuous on $\mathbb{R}^{N}$ in the second sense, hence so is $-\beta^{-1}(\Theta\circ f)\,\partial_{i}f$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, and so it is continuous at every point in the first sense. Hence $\Theta\circ f$ is of class $C^{1}$ by clause 1 of C^k Maps on a Euclidean Open Set, read through its clause 3.

(d) Cylindrical functions. Let $F,\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $t\in\mathbb{R}$. By (b) they have representations $(N,\psi_{1})$, $(N,\psi_{2})$ with a common $N$; by (a), $(N,\psi_{1}\psi_{2})$ and $(N,\psi_{1}+t\psi_{2})$ are representations of $F\varphi$ and $F+t\varphi$, which therefore belong to $\mathcal{F}C^{1}_{b}(X)$ (Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical); and by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial and (a), for every $k\in\mathbb{N}$

$$\partial_{k}(F\varphi)=\partial_{k}F\,\varphi+F\,\partial_{k}\varphi,\qquad\partial_{k}(F+t\varphi)=\partial_{k}F+t\,\partial_{k}\varphi,$$

all terms vanishing for $k>N$. A constant function belongs to $\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §linear; explicitly, the function with constant value $b'\in\mathbb{R}$ has the representation $(1,\psi_{b'})$, where $\psi_{b'}:\mathbb{R}^{1}\to\mathbb{R}$ is the constant function with value $b'$, which is smooth, hence of class $C^{1}$, by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and Smooth Map on a Euclidean Open Set, and whose partial derivative vanishes by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives and claim 2 of Slice Function and the Partial Derivative, its slices being constant; so $\psi_{b'}\in C^{1}_{b}(\mathbb{R}^{1})$, and the partial derivatives of the constant function vanish by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial.

Step 2 (Integration by parts against the Gibbs measure). For $u\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$ put

$$r_{u,k}(x)=\Bigl(\frac{x_{k}}{c_{k}}+\frac{\partial_{k}V(x)}{\beta}\Bigr)u(x)-\partial_{k}u(x)\qquad(x\in X).$$

We show: $r_{u,k}$ is integrable with respect to $\gamma^{V}$ and $\int_{X}r_{u,k}\,d\gamma^{V}=0$. It is Borel: $x\mapsto x_{k}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, $u$ and $\partial_{k}u$ 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, $\partial_{k}V$ as noted, and claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions apply ($c_{k}>0$ by Variance Sequences and Their Truncations §variances).

Let $(n_{u},\chi)$ be a representation of $u$ and $N=\max\{n_{u},d\}$. By Step 1(b), $u=\tilde{\chi}\circ p_{N}$ with $\tilde{\chi}=\chi\circ\pi\in C^{1}_{b}(\mathbb{R}^{N})$ ($\pi$ the projection onto the first $n_{u}$ coordinates), and, $v$ being of class $C^{2}$ and so of class $C^{1}$ (clause 2 of C^k Maps on a Euclidean Open Set), $V=\tilde{v}\circ p_{N}$ with $\tilde{v}=v\circ\pi'$ of class $C^{1}$ ($\pi'$ the projection onto the first $d$ coordinates), $\partial_{i}\tilde{v}=(\partial_{i}v)\circ\pi'$ for $i\le d$ and $\partial_{i}\tilde{v}=0$ for $d<i\le N$. Comparing with Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient, $(\partial_{k}\tilde{v})\circ p_{N}=\partial_{k}V$ for $k\le N$, and $\partial_{k}V=0$ for $k>N$. Put $\psi=\tilde{\chi}\,(\Theta\circ\tilde{v})$. By Step 1(a),(c), $\psi$ is of class $C^{1}$ on $\mathbb{R}^{N}$ with

$$\partial_{i}\psi=\Bigl(\partial_{i}\tilde{\chi}-\frac{1}{\beta}\,\tilde{\chi}\,\partial_{i}\tilde{v}\Bigr)(\Theta\circ\tilde{v}),$$

so that, by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial for the representation $(N,\tilde{\chi})$ of $u$,

$$\psi\circ p_{N}=u\,w,\qquad(\partial_{k}\psi)\circ p_{N}=\Bigl(\partial_{k}u-\frac{1}{\beta}\,u\,\partial_{k}V\Bigr)w\quad(k\le N).$$

Growth bound. Let $B\ge0$ bound $|\tilde{\chi}|$ and every $|\partial_{i}\tilde{\chi}|$, and let $A=\max_{i\in[d]}(C/a_{i})^{1/2}$. Let $y\in\mathbb{R}^{N}$ and $t=\tilde{v}(y)=v(\pi'(y))\ge-b$. For $i\le d$, the slope bound gives $a_{i}(\partial_{i}\tilde{v}(y))^{2}\le\sum_{l=1}^{d}a_{l}(\partial_{l}v(\pi'(y)))^{2}\le C(1+|t|)^{2}$, so $|\partial_{i}\tilde{v}(y)|\le A(1+|t|)$; for $i>d$, $\partial_{i}\tilde{v}(y)=0$. Put $s=t+b\ge0$; then $1+|t|\le1+|b|+s$ (if $t\ge0$ then $|t|=t\le t+b+|b|$; if $t<0$ then $|t|=-t\le b$). By claims 1, 2 and 4 of Basic Properties of the Exponential Function, $\Theta(t)=\exp(b/\beta)\exp(-s/\beta)$, $\exp(-s/\beta)=1/\exp(s/\beta)$, $\Theta(t)\le\exp(b/\beta)$, and by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §exp, $\exp(s/\beta)\ge1+s/\beta>0$. Hence

$$(1+|t|)\,\Theta(t)\le\exp(b/\beta)\,\frac{(1+|b|+s)\,\beta}{\beta+s}\le\exp(b/\beta)\,(1+|b|+\beta),$$

since $\beta/(\beta+s)\le1$ and $s\beta/(\beta+s)\le\beta$. Therefore, for all $y$ and $i\in[N]$,

$$|\partial_{i}\psi(y)|\le M:=B\exp(b/\beta)\Bigl(1+\frac{A}{\beta}(1+|b|+\beta)\Bigr)\le M(1+\lVert y\rVert).$$

Integration by parts. By Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space with $n=N$, this $\psi$ and this $M$ (the functions written $\varphi$ and $\varphi_{i}$ there being $uw$ and $(\partial_{i}\psi)\circ p_{N}$), the functions $uw$, $(\partial_{i}\psi)\circ p_{N}$ and $x\mapsto x_{k}u(x)w(x)$ are integrable with respect to $\gamma_{c}$ (Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space §integrable). If $k\le N$, then $r_{u,k}w=c_{k}^{-1}x_{k}uw-(\partial_{k}\psi)\circ p_{N}$ is integrable by linearity, and its integral is $0$ by Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space §coordinate. If $k>N$, then $\partial_{k}u=0$ (Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial) and $\partial_{k}V=0$, so $r_{u,k}w=c_{k}^{-1}x_{k}uw$, integrable with integral $0$ by Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space §orthogonal. In both cases Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §density shows that $r_{u,k}$ is integrable with respect to $\gamma^{V}$ and

$$\int_{X}r_{u,k}\,d\gamma^{V}=\frac{1}{Z_{V,\beta}}\int_{X}r_{u,k}\,w\,d\gamma_{c}=0. \tag{2.1}$$

Step 3 (The Gibbs entropy pair). Since $c_{k}\le\kappa a_{k}$ for all $k$, the hypothesis of the Gibbs entropy pair holds with this $\kappa$; let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gibbs entropy pair with potential $V$ and temperature $\beta$. It is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gibbs Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair, and it is $\lambda$-displacement convex by The Gibbs Entropy Pair of a K-Semiconvex Potential is Lambda-Displacement Convex with Lambda the Temperature over the Variance-to-Noise Bound Minus K §convex, $K$ being nonnegative and the semiconvexity constant of $V$. We record:

(F1) For $\mu\in\mathcal{D}$: $\mu$ has finite relative entropy with respect to $\gamma_{c}$ and $V$ is integrable with respect to $\mu$ by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §entropy; $\mu\in\mathcal{P}_{2}(X)$ and each $\partial_{k}V$ is integrable with respect to $\mu$ by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §domain; and $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain.

(F2) For $\mu\in\mathcal{D}_{\Sigma}$: by (F1) and Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §splitting, $\mu$ has a relative score with respect to $\gamma_{c}$, finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$; so Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §field applies and, with $\Sigma(\mu)=\beta Z^{a}_{\mu}+\nabla_{a}V$ (The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §pair),

$$\lVert\Sigma(\mu)\rVert_{\mu}^{2}=\beta^{2}\,\mathcal{I}_{a}(\mu\,|\,\gamma^{V}). \tag{3.1}$$

Step 4 (The Gibbs measure is a zero of the score). The constant function $1$ is a density of $\gamma^{V}$ with respect to itself, since $\int_{X}\mathbf{1}_{A}\cdot1\,d\gamma^{V}=\gamma^{V}(A)$ for $A\in\mathcal{B}(X)$ (integral of an indicator); and $\phi(1)=\log1=\log\exp(0)=0$ by The Natural Logarithm and claim 1 of Basic Properties of the Exponential Function. So $\phi\circ1$ is the zero function, integrable with integral $0$: $\gamma^{V}$ has finite relative entropy with respect to itself (Relative Entropy of Probability Measures §relative-entropy), $\gamma^{V}\in\mathcal{D}$, $H(\gamma^{V}\,|\,\gamma^{V})=0$ and $\mathcal{E}(\gamma^{V})=0$.

By (F1), $\gamma^{V}\in\mathcal{P}_{2}(X)$ and every $\partial_{k}V$ is integrable with respect to $\gamma^{V}$, so The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential applies to $\gamma^{V}$. For $k\in\mathbb{N}$ let $\zeta^{V}_{k}$ be the zero vector of $L^{2}(\gamma^{V})$. For every $\varphi\in\mathcal{F}C^{1}_{b}(X)$, $\langle\zeta^{V}_{k},\varphi\rangle_{L^{2}(\gamma^{V})}=0$ by Elementary Identities in a Real Inner Product Space §zero, and $\int_{X}r_{\varphi,k}\,d\gamma^{V}=0$ by (2.1). So $(\zeta^{V}_{k})_{k\in\mathbb{N}}$ is the relative score of $\gamma^{V}$ with respect to $\gamma^{V}$ (The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential §score); the series $\sum_{k}a_{k}\lVert\zeta^{V}_{k}\rVert^{2}$ has all terms $0$, so it converges with sum $0$ (Series of Real Numbers §convergent), and $\gamma^{V}$ has finite Fisher information relative to $\gamma^{V}$ with weights $a$, with $\mathcal{I}_{a}(\gamma^{V}\,|\,\gamma^{V})=0$ (The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential §information). Hence $\gamma^{V}\in\mathcal{D}_{\Sigma}$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain, and by (3.1) $\lVert\Sigma(\gamma^{V})\rVert_{\gamma^{V}}=0$. As $L^{2}(\gamma^{V};X^{a})$ is a real Hilbert space with this norm (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert), $\Sigma(\gamma^{V})$ is its zero vector by Elementary Identities in a Real Inner Product Space §vanishing. Thus Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score applies to the pair, this $\lambda$, and $\mu_{*}=\gamma^{V}$, with $\mathcal{E}(\mu_{*})=0$.

Step 5 (Claim 1). Let $\mu$ be as in claim 1. Then $\mu\in\mathcal{D}$ and, by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain, $\mu\in\mathcal{D}_{\Sigma}$. By Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score §log-sobolev and (3.1),

$$\beta\,H(\mu\,|\,\gamma^{V})=\mathcal{E}(\mu)-\mathcal{E}(\gamma^{V})\le\frac{1}{2\lambda}\,\lVert\Sigma(\mu)\rVert_{\mu}^{2}=\frac{\beta^{2}}{2\lambda}\,\mathcal{I}_{a}(\mu\,|\,\gamma^{V}).$$

Dividing by $\beta>0$ and using $1/\lambda=\kappa/(\beta-\kappa K)$ gives $H(\mu\,|\,\gamma^{V})\le\frac{\kappa\beta}{2(\beta-\kappa K)}\,\mathcal{I}_{a}(\mu\,|\,\gamma^{V})$.

Step 6 (Claim 2). Let $\mu$ have finite relative entropy with respect to $\gamma^{V}$. Then $\mu\in\mathcal{D}$, and $\gamma^{V}\in\mathcal{D}$ by Step 4; both lie in $\mathcal{P}^{a}_{\rho}$ by (F1). By Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score §talagrand with $\nu=\mu$, and 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 §symmetry,

$$\frac{\lambda}{2}\,W_{a}(\mu,\gamma^{V})^{2}=0+\frac{\lambda}{2}\,W_{a}(\gamma^{V},\mu)^{2}\le\mathcal{E}(\mu)=\beta\,H(\mu\,|\,\gamma^{V}),$$

so $W_{a}(\mu,\gamma^{V})^{2}\le\frac{2\beta}{\lambda}H(\mu\,|\,\gamma^{V})=\frac{2\kappa\beta}{\beta-\kappa K}H(\mu\,|\,\gamma^{V})$.

Step 7 (A bound for $\phi$). Let $L\ge0$ and let $G:X\to[0,L]$ be Borel. Then $\phi\circ G$ 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, $-e^{-1}\le\phi\circ G$ with $e^{-1}=\exp(-1)$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §lower, and for $0<s\le L$, $\phi(s)=s\log s\le s(s-1)\le L^{2}$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log, while $\phi(0)=0$. So $|\phi\circ G|\le e^{-1}+L^{2}$, and $G$, $\phi\circ G$ are bounded Borel functions, integrable with respect to every Borel probability measure on $X$. In particular, for $F\in\mathcal{F}C^{1}_{b}(X)$, which is Borel with $|F|\le B$ for some $B\ge0$ 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, the functions $F^{2}$ and $\phi\circ F^{2}$ are bounded and Borel (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), hence integrable with respect to $\gamma^{V}$; this is the first assertion of claim 3.

Step 8 (Claim 3 for $F^{2}+\varepsilon$). Let $F\in\mathcal{F}C^{1}_{b}(X)$ with representation $(n,\psi_{F})$, $|F|\le B$ and $|\partial_{k}F|\le B_{k}$ (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 fix a positive $\varepsilon\in\mathbb{R}$. By Step 1(d) applied to $F$ and $F$, $F^{2}\in\mathcal{F}C^{1}_{b}(X)$ with $\partial_{k}(F^{2})=2F\,\partial_{k}F$; by Step 1(d) applied to $F^{2}$ and the constant function $1$, with $t=\varepsilon$, $g=F^{2}+\varepsilon\in\mathcal{F}C^{1}_{b}(X)$ with $\partial_{k}g=2F\,\partial_{k}F$ for every $k$ (so $\partial_{k}g=0$ for $k>n$, by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial for $F$), and $\varepsilon\le g\le B^{2}+\varepsilon$. Following the construction of Step 1(d), with the representations $(n,\psi_{F})$ of $F$ and $(1,\psi_{b'})$ of $1$ (Step 1(d) with $b'=1$) lifted by Step 1(b) to a common dimension $N\ge n$, $g$ has a representation $(N,\psi_{g})$ with $\psi_{g}=\tilde{\psi}_{F}^{2}+\varepsilon$, where $\tilde{\psi}_{F}=\psi_{F}\circ\pi$ and $\pi$ is the projection onto the first $n$ coordinates; in particular $\psi_{g}(y)\ge\varepsilon>0$ for every $y\in\mathbb{R}^{N}$.

(a) The measure. Put $m=\int_{X}g\,d\gamma^{V}$; by monotonicity and claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space, $\varepsilon\le m$. Let $h=g/m$, a Borel function with $\varepsilon/m\le h\le(B^{2}+\varepsilon)/\varepsilon$ and $\int_{X}h\,d\gamma^{V}=1$. By claim 3 of Image Measures, Measures with Densities, and Change of Variables, $\mu(A)=\int_{X}\mathbf{1}_{A}h\,d\gamma^{V}$ defines a measure $\mu$ on $\mathcal{B}(X)$ with $\mu(X)=1$, so $\mu\in\mathcal{P}(X)$, and for every Borel $f$, $f$ is integrable with respect to $\mu$ if and only if $fh$ is integrable with respect to $\gamma^{V}$, in which case $\int_{X}f\,d\mu=\int_{X}fh\,d\gamma^{V}$. Thus $h$ is a density of $\mu$ with respect to $\gamma^{V}$, $\phi\circ h$ is integrable by Step 7, and $\mu$ has finite relative entropy with respect to $\gamma^{V}$ (Relative Entropy of Probability Measures §relative-entropy); so $\mu\in\mathcal{D}$.

(b) The entropy. Since $g=m\,h$ with $m,h$ positive, $\log g=\log m+\log h$ by The Natural Logarithm, so pointwise $\phi(h)=h\log h=m^{-1}\phi(g)-m^{-1}(\log m)\,g$. By Step 7, $g$ and $\phi\circ g$ are integrable with respect to $\gamma^{V}$, so $\operatorname{Ent}_{\gamma^{V}}(g)$ is defined (The Entropy of a Nonnegative Function with Respect to a Probability Measure §entropy), and by linearity

$$H(\mu\,|\,\gamma^{V})=\int_{X}\phi\circ h\,d\gamma^{V}=\frac{1}{m}\Bigl(\int_{X}\phi\circ g\,d\gamma^{V}-m\log m\Bigr)=\frac{1}{m}\operatorname{Ent}_{\gamma^{V}}(g).$$

(c) The relative score. By (F1), $\mu\in\mathcal{P}_{2}(X)$ and every $\partial_{k}V$ is integrable with respect to $\mu$, so The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential applies to $\mu$. The function $\psi_{g}$ is of class $C^{1}$ on $\mathbb{R}^{N}$, hence continuous on $(\mathbb{R}^{N},d_{E})$ by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and nowhere zero, so $1/\psi_{g}$ is continuous on $\mathbb{R}^{N}$ by claim 2 of Continuity of the Reciprocal of a Nonvanishing Real-Valued Function on a Metric Space and Borel by claims 2 and 3 of Borel Measurability and Bounded Integration on a Metric Space; as $p_{N}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, $1/g=(1/\psi_{g})\circ p_{N}$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. For $k\in\mathbb{N}$ let $\zeta_{k}=\partial_{k}g\cdot(1/g)$, which 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 and claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, with $|\zeta_{k}|\le2BB_{k}/\varepsilon$ ($\zeta_{k}=0$ for $k>n$), so that $\zeta_{k}^{2}$ is bounded and Borel, $\zeta_{k}$ is $2$-integrable with respect to $\mu$, and its class lies in $L^{2}(\mu)$. Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$. By Step 1(d), $g\varphi\in\mathcal{F}C^{1}_{b}(X)$ with $\partial_{k}(g\varphi)=\partial_{k}g\,\varphi+g\,\partial_{k}\varphi$, so pointwise

$$g\,r_{\varphi,k}=\Bigl(\frac{x_{k}}{c_{k}}+\frac{\partial_{k}V}{\beta}\Bigr)g\varphi-g\,\partial_{k}\varphi=r_{g\varphi,k}+\varphi\,\partial_{k}g .$$

By Step 2, $r_{g\varphi,k}$ is integrable with respect to $\gamma^{V}$ with integral $0$, and $\varphi\,\partial_{k}g$ is bounded and Borel; so $h\,r_{\varphi,k}=m^{-1}g\,r_{\varphi,k}$ is integrable with respect to $\gamma^{V}$, and by (a) and linearity

$$\int_{X}r_{\varphi,k}\,d\mu=\int_{X}r_{\varphi,k}\,h\,d\gamma^{V}=\frac{1}{m}\int_{X}\varphi\,\partial_{k}g\,d\gamma^{V}=\int_{X}\varphi\,\zeta_{k}\,h\,d\gamma^{V}=\int_{X}\zeta_{k}\,\varphi\,d\mu=\langle\zeta_{k},\varphi\rangle_{L^{2}(\mu)},$$

using $\zeta_{k}h=m^{-1}\partial_{k}g$, the bounded Borel function $\zeta_{k}\varphi$, and The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product. By The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential §score, $(\zeta_{k})_{k\in\mathbb{N}}$ is the relative score of $\mu$ with respect to $\gamma^{V}$.

(d) The Fisher information. By The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and (a), $\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}=\int_{X}\zeta_{k}^{2}h\,d\gamma^{V}=m^{-1}\int_{X}(\partial_{k}g)^{2}/g\,d\gamma^{V}$, the integrand $(\partial_{k}g)^{2}/g=\zeta_{k}^{2}\,g$ being a product of bounded Borel functions (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); this is $0$ for $k>n$. So the series $\sum_{k}a_{k}\lVert\zeta_{k}\rVert^{2}$ converges (its partial sums are constant from the $n$-th on), $\mu$ has finite Fisher information relative to $\gamma^{V}$ with weights $a$ (The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential §information), and, since $(\partial_{k}g)^{2}/g=4F^{2}(\partial_{k}F)^{2}/(F^{2}+\varepsilon)\le4(\partial_{k}F)^{2}$, monotonicity and The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient give

$$\mathcal{I}_{a}(\mu\,|\,\gamma^{V})=\frac{1}{m}\int_{X}\sum_{k=1}^{n}a_{k}\frac{(\partial_{k}g)^{2}}{g}\,d\gamma^{V}\le\frac{4}{m}\int_{X}\sum_{k=1}^{n}a_{k}(\partial_{k}F)^{2}\,d\gamma^{V}=\frac{4}{m}\int_{X}|\nabla_{a}F|_{a}^{2}\,d\gamma^{V},$$

the last integrand being bounded and Borel by that clause.

(e) Conclusion. By (a), (c) and (d), $\mu$ satisfies the hypotheses of claim 1, so by Step 5, (b) and (d)

$$\frac{1}{m}\operatorname{Ent}_{\gamma^{V}}(F^{2}+\varepsilon)\le\frac{\kappa\beta}{2(\beta-\kappa K)}\cdot\frac{4}{m}\int_{X}|\nabla_{a}F|_{a}^{2}\,d\gamma^{V},$$

and multiplying by $m>0$,

$$\operatorname{Ent}_{\gamma^{V}}(F^{2}+\varepsilon)\le\frac{2\kappa\beta}{\beta-\kappa K}\int_{X}|\nabla_{a}F|_{a}^{2}\,d\gamma^{V}. \tag{8.1}$$

Step 9 (Claim 3). Apply (8.1) with $\varepsilon=1/j$, $j\in\mathbb{N}$. For each $x\in X$, $F(x)^{2}+1/j\to F(x)^{2}$, so $\phi(F(x)^{2}+1/j)\to\phi(F(x)^{2})$ by 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, unwound through the $\varepsilon$-$\delta$ condition of Continuous Map Between Metric Spaces); and $|\phi\circ(F^{2}+1/j)|\le e^{-1}+(B^{2}+1)^{2}$ by Step 7 with $L=B^{2}+1$, a constant, integrable with respect to $\gamma^{V}$. By Dominated Convergence Theorem, $\int_{X}\phi\circ(F^{2}+1/j)\,d\gamma^{V}\to\int_{X}\phi\circ F^{2}\,d\gamma^{V}$. Also $\int_{X}(F^{2}+1/j)\,d\gamma^{V}=\int_{X}F^{2}\,d\gamma^{V}+1/j\to\int_{X}F^{2}\,d\gamma^{V}$ by linearity and claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space, so $\phi\bigl(\int_{X}(F^{2}+1/j)\,d\gamma^{V}\bigr)\to\phi\bigl(\int_{X}F^{2}\,d\gamma^{V}\bigr)$ by continuity of $\phi$. Hence, by The Entropy of a Nonnegative Function with Respect to a Probability Measure §entropy, $\operatorname{Ent}_{\gamma^{V}}(F^{2}+1/j)\to\operatorname{Ent}_{\gamma^{V}}(F^{2})$, and letting $j\to\infty$ in (8.1), whose right-hand side does not depend on $j$,

$$\operatorname{Ent}_{\gamma^{V}_{\beta}}(F^{2})\le\frac{2\kappa\beta}{\beta-\kappa K}\int_{X}|\nabla_{a}F|_{a}^{2}\,d\gamma^{V}_{\beta}.$$

This proves claim 3.