Proof of The Gibbs Entropy Pair Has Regular Penalised Maxima

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

Real arithmetic, the order of $\mathbb{R}$ and absolute values are those of The Real Numbers: Standing Notation and Background §background, used without further mention; square roots are the nonnegative ones of Existence and Uniqueness of the Nonnegative Square Root, and for nonnegative reals $s,t$, $s\le t$ if and only if $s^{2}\le t^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

The quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ 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, 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

$$\chi(\nu)-\chi(\mu)\le\lambda\bigl(\mathcal{E}(\nu)-\mathcal{E}(\mu)\bigr).\tag{0.1}$$

We must show $\mu\in\mathcal{D}_{\Sigma}$. 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.

Data on $V$ and on $\mu$. Write $V=v\circ p_{d}$ with head dimension $d$, profile $v$ and semiconvexity constant $K\ge0$ (Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §admissible), fix $b$ 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, so $C\ge0$, and let $L=C^{1/2}(|b+1|+2)\ge0$ be the number of Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §translation. Let $Z_{V,\beta}$ be the normaliser of The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §normaliser. Since $\mu\in\mathcal{D}$, $\mu$ has finite relative entropy with respect to $\gamma^{V}_{\beta}$ (The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain); by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §entropy, $\mu$ has finite relative entropy with respect to $\gamma_{c}$ and $V$ is integrable with respect to $\mu$, and by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §domain, $\mu\in\mathcal{P}_{2}(X)$ and each $\partial_{k}V$ is integrable with respect to $\mu$. For every $\nu\in\mathcal{D}$, The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §pair and Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §entropy give

$$\mathcal{E}(\nu)=\beta H(\nu\,|\,\gamma^{V}_{\beta})=\beta H(\nu\,|\,\gamma_{c})+\int_{X}V\,d\nu+\beta\log Z_{V,\beta}.\tag{0.2}$$

For every $y\in X$ we have $-b\le V(y)$ and $V(y)+b+1\ge1$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, hence $|V(y)|\le V(y)+b+1+|b|$ (if $V(y)\ge0$ this holds as $b+|b|+1\ge0$; if $V(y)<0$, then $|V(y)|=-V(y)\le b\le|b|\le V(y)+b+1+|b|$, as $V(y)+b+1\ge1$); and by Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §slope and Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, $|\nabla_{a}V(y)|_{a}^{2}=\sum_{k=1}^{d}a_{k}(\partial_{k}v(p_{d}(y)))^{2}\le C(1+|V(y)|)^{2}$, so

$$|\nabla_{a}V(y)|_{a}\le C^{1/2}\bigl(1+|V(y)|\bigr)\le C^{1/2}\bigl(V(y)+b+2+|b|\bigr)\qquad(y\in X).\tag{0.3}$$

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 with noise gradient $\nabla_{a}\psi(x)=\sum_{k=1}^{n}a_{k}\,\partial_{k}g(p_{n}(x))\,e_{k}$, by the preamble of Relative Entropy along a Noise-Gradient Perturbation of the Identity: the Formula, the Second-Order Expansion and the First Variation (whose hypothesis holds for $\mu$); so the $k$-th coordinate of $\nabla_{a}\psi(x)$ is $a_{k}\partial_{k}g(p_{n}(x))$ for $k\le n$ and $0$ for $k>n$. Let $h\in L^{2}(\mu;X^{a})$ be the class of $\nabla_{a}\psi$, 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 A Bound on the Gibbs Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Fisher Information Relative to the Gibbs Measure. By Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded 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 $i,j,k\in[n]$; these are the numbers written $B$ and $b$ in Relative Entropy along a Noise-Gradient Perturbation of the Identity: the Formula, the Second-Order Expansion and the First Variation. Let $\theta_{n}>0$ and $K_{n}\ge0$ be the constants of that lemma (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 $c_{d}$ and $K_{d}$), and put

$$C_{g}=K_{n}b'^{2}+B^{2}\sum_{k=1}^{n}\frac{a_{k}^{2}}{2c_{k}}\ge0,\quad\ell_{0}=L^{a}_{\mu}(g),\quad G=\langle\nabla\chi(\mu),h\rangle_{\mu},\quad B_{\psi}=\Bigl(\sum_{k=1}^{n}a_{k}B^{2}\Bigr)^{1/2},\quad t_{*}=(1+2nb')^{-1},$$

where $L^{a}_{\mu}(g)$ is the noise Ornstein-Uhlenbeck functional. By the formula of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, $|\nabla_{a}\psi(x)|_{a}^{2}=\sum_{k=1}^{n}a_{k}(\partial_{k}g(p_{n}(x)))^{2}\le B_{\psi}^{2}$, so $|\nabla_{a}\psi(x)|_{a}\le B_{\psi}$ for every $x$. Note $0<t_{*}\le1$, and every $t$ with $|t|\le t_{*}$ satisfies $2n|t|b'\le2nb'(1+2nb')^{-1}\le1$. For $t\in\mathbb{R}$ let $S_{t}(x)=x+t\,\nabla_{a}\psi(x)$, the map $\mathrm{id}+t\nabla_{a}\psi$ of Relative Entropy along a Noise-Gradient Perturbation of the Identity: the Formula, the Second-Order Expansion and the First Variation, 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 $\mathcal{J}^{a}(\nabla\chi(\mu),\pi_{t})=t\,G$.

The potential pairing. By Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, applied with the vector $\nabla_{a}\psi(x)\in X^{a}$, and since $\partial_{k}V=0$ for $k>d$ (Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient), for all $t\in\mathbb{R}$ and $x\in X$ the function

$$q_{t}(x)=\langle\nabla_{a}V(S_{t}(x)),\nabla_{a}\psi(x)\rangle_{a}=\sum_{k=1}^{n}a_{k}\,\partial_{k}V(S_{t}(x))\,\partial_{k}g(p_{n}(x))\tag{1.1}$$

is given by the displayed finite sum. Each product $\partial_{k}V\,(\partial_{k}g\circ p_{n})$ is integrable with respect to $\mu$ by the preamble of The Gibbs Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates, so $q_{0}$ is integrable; put $P=\int_{X}q_{0}\,d\mu$. For each $k\in[n]$ the integrand of the $k$-th term of The Gibbs Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional is that of the $k$-th term of The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional plus $\beta^{-1}\partial_{k}V\,(\partial_{k}g\circ p_{n})$, so the linearity of Linearity and Monotonicity of the Lebesgue Integral §integrable and (1.1) with $t=0$ give

$$L^{a,V}_{\mu}(g)=\ell_{0}+\beta^{-1}P,\tag{1.2}$$

the left-hand side being the Gibbs Ornstein-Uhlenbeck functional, defined as $\mu\in\mathcal{P}_{2}(X)$ and $V$ is integrable with respect to $\mu$.

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 (The perturbed measures lie in $\mathcal{D}$, and the change of the potential energy). Let $t\in\mathbb{R}$ with $|t|\le t_{*}$, so $|t|\le1$ and $2n|t|b'\le1$. By Relative Entropy along a Noise-Gradient Perturbation of the Identity: the Formula, the Second-Order Expansion and the First Variation §pushforward, $S_{t}$ is Borel and $\mu_{t}$ has finite relative entropy with respect to $\gamma_{c}$. Let $x\in X$. The vector $t\nabla_{a}\psi(x)$ lies in $X^{a}$, a linear subspace of $X$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert, and $|t\nabla_{a}\psi(x)|_{a}=|t|\,|\nabla_{a}\psi(x)|_{a}\le B_{\psi}$. So Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §translation, the positivity $V(x)+b+1\ge1$ and the monotonicity of $\exp$ (claim 4 of Basic Properties of the Exponential Function) give

$$1\le V(S_{t}(x))+b+1\le\bigl(V(x)+b+1\bigr)\exp(LB_{\psi}),$$

and therefore, by the bound $|V(y)|\le V(y)+b+1+|b|$ of the data paragraph,

$$1+|V(S_{t}(x))|\le\Lambda(x):=\bigl(V(x)+b+1\bigr)\exp(LB_{\psi})+|b|+1 .\tag{3.1}$$

The function $\Lambda$ is integrable with respect to $\mu$ by Linearity and Monotonicity of the Lebesgue Integral §integrable, $V$ being integrable and constants integrable by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space. The function $V\circ S_{t}$ is Borel, by claims 3 and 4 of Borel Measurability and Bounded Integration on a Metric Space, $V$ being continuous (Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity); and $|V\circ S_{t}|\le\Lambda$, so $V\circ S_{t}$ is integrable, by the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative and the definition of integrability. By the change of variables formula of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward (claim 2 of Image Measures, Measures with Densities, and Change of Variables), $V$ is integrable with respect to $\mu_{t}$ and $\int_{X}V\,d\mu_{t}=\int_{X}V\circ S_{t}\,d\mu$. Hence, by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §entropy, $\mu_{t}$ has finite relative entropy with respect to $\gamma^{V}_{\beta}$, that is, $\mu_{t}\in\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ (The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain), and (0.2) for $\nu=\mu_{t}$ and $\nu=\mu$, with the linearity of Linearity and Monotonicity of the Lebesgue Integral §integrable, gives

$$\mathcal{E}(\mu_{t})-\mathcal{E}(\mu)=\beta\bigl(H(\mu_{t}\,|\,\gamma_{c})-H(\mu\,|\,\gamma_{c})\bigr)+\int_{X}\bigl(V\circ S_{t}-V\bigr)\,d\mu.\tag{3.2}$$

Next, $x-S_{t}(x)=-t\nabla_{a}\psi(x)\in X^{a}$, so Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §tangent-inequality, applied at the points $S_{t}(x)$ and $x$ (in the roles of its $x$ and $y$), gives $V(x)\ge V(S_{t}(x))-t\,q_{t}(x)-\tfrac{K}{2}t^{2}|\nabla_{a}\psi(x)|_{a}^{2}$, with $q_{t}$ of (1.1); since $K\ge0$,

$$V(S_{t}(x))-V(x)\le t\,q_{t}(x)+\tfrac{K}{2}\,t^{2}B_{\psi}^{2}.\tag{3.3}$$

The function $q_{t}$ is Borel by (1.1), each $\partial_{k}V$ being continuous (Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity) and each $\partial_{k}g\circ p_{n}$ Borel (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), with claims 3 and 4 of Borel Measurability and Bounded Integration on a Metric Space and claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By the Cauchy-Schwarz inequality in the real Hilbert space $X^{a}$ (The Cauchy-Schwarz Inequality in a Real Inner Product Space, The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert), (0.3) and (3.1),

$$|q_{t}(x)|\le|\nabla_{a}V(S_{t}(x))|_{a}\,|\nabla_{a}\psi(x)|_{a}\le C^{1/2}B_{\psi}\,\Lambda(x),\tag{3.4}$$

so $q_{t}$ is integrable. Integrating (3.3), by the linearity and monotonicity of Linearity and Monotonicity of the Lebesgue Integral §integrable,

$$\int_{X}\bigl(V\circ S_{t}-V\bigr)\,d\mu\le t\int_{X}q_{t}\,d\mu+\tfrac{K}{2}\,t^{2}B_{\psi}^{2}\qquad(|t|\le t_{*}).\tag{3.5}$$

Step 4 (Continuity of the potential pairing at $t=0$). We claim: for every positive $\eta\in\mathbb{R}$ there is $t_{\eta}\in\mathbb{R}$ with $0<t_{\eta}\le t_{*}$ such that $\bigl|\int_{X}q_{t}\,d\mu-P\bigr|\le\eta$ whenever $|t|\le t_{\eta}$. Suppose not, for some $\eta>0$. Then for each $j\in\mathbb{N}$ the number $t_{*}/j$ does not qualify, so there is $t_{j}$ with $|t_{j}|\le t_{*}/j$ and $\bigl|\int_{X}q_{t_{j}}\,d\mu-P\bigr|>\eta$. Then $t_{j}\to0$ by The Archimedean Property of the Real Numbers and Limit of a Sequence of Real Numbers. Fix $x\in X$ and $k\in[d]$. The distance from $S_{t_{j}}(x)$ to $x$ in $X$ is $|t_{j}|\,|\nabla_{a}\psi(x)|$, which tends to $0$; given $\varepsilon'>0$, the continuity of $\partial_{k}V$ at $x$ (Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, in the sense of Continuous Map Between Metric Spaces) provides $\delta>0$ with $|\partial_{k}V(y)-\partial_{k}V(x)|<\varepsilon'$ whenever $|y-x|<\delta$, and $|t_{j}|\,|\nabla_{a}\psi(x)|<\delta$ for all large $j$; so $\partial_{k}V(S_{t_{j}}(x))\to\partial_{k}V(x)$, and $q_{t_{j}}(x)\to q_{0}(x)$ by (1.1) and Arithmetic of Limits of Real Sequences. Since $|t_{j}|\le t_{*}$, (3.4) gives $|q_{t_{j}}|\le C^{1/2}B_{\psi}\Lambda$, an integrable function. By claim 3 of Dominated Convergence Theorem (on the measure space $(X,\mathcal{B}(X),\mu)$, with the Borel functions $q_{t_{j}}$ and limit function $q_{0}$), $\int_{X}q_{t_{j}}\,d\mu\to\int_{X}q_{0}\,d\mu=P$, so $\bigl|\int_{X}q_{t_{j}}\,d\mu-P\bigr|<\eta$ for all large $j$ (Limit of a Sequence of Real Numbers), a contradiction. This proves the claim.

Step 5 ($\lambda\beta L^{a,V}_{\mu}(g)=G$). 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 $I^{a}(\pi)<\theta_{\chi}^{2}$. Next let $t_{\varepsilon/\lambda}$ be given by Step 4 with $\eta=\varepsilon/\lambda$. Then choose a positive $s\in\mathbb{R}$ with

$$s\le t_{\varepsilon/\lambda},\qquad s\,b'\le\theta_{n},\qquad s\lVert h\rVert_{\mu}<\theta_{\chi},\qquad s\lVert h\rVert_{\mu}<r,\qquad\lambda\beta C_{g}\,s\le\varepsilon,\qquad\lambda KB_{\psi}^{2}\,s\le2\varepsilon,$$

which is possible since each condition holds for all sufficiently small positive $s$. Let $t\in\{s,-s\}$, so $|t|=s\le t_{*}$ and $2n|t|b'\le1$.

By Step 3, $\mu_{t}\in\mathcal{D}$. 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. Hence (0.1) gives $\chi(\mu_{t})-\chi(\mu)\le\lambda(\mathcal{E}(\mu_{t})-\mathcal{E}(\mu))$.

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 $H(\mu_{t}\,|\,\gamma_{c})-H(\mu\,|\,\gamma_{c})\le t\ell_{0}+C_{g}s^{2}$. Since $|t|=s\le t_{\varepsilon/\lambda}$, Step 4 gives $t\int_{X}q_{t}\,d\mu=tP+t\bigl(\int_{X}q_{t}\,d\mu-P\bigr)\le tP+s\varepsilon/\lambda$. With (3.2) and (3.5),

$$\mathcal{E}(\mu_{t})-\mathcal{E}(\mu)\le\beta t\ell_{0}+\beta C_{g}s^{2}+tP+\frac{s\varepsilon}{\lambda}+\frac{K}{2}B_{\psi}^{2}s^{2}.$$

Multiplying by $\lambda>0$ and using $\lambda\beta C_{g}s\le\varepsilon$ and $\lambda\tfrac{K}{2}B_{\psi}^{2}s\le\varepsilon$, $\lambda(\mathcal{E}(\mu_{t})-\mathcal{E}(\mu))\le\lambda t(\beta\ell_{0}+P)+3\varepsilon s$. Combining the three inequalities,

$$tG-\varepsilon s\lVert h\rVert_{\mu}\le\lambda t(\beta\ell_{0}+P)+3\varepsilon s,\qquad\text{that is}\qquad t\bigl(G-\lambda(\beta\ell_{0}+P)\bigr)\le\varepsilon s\bigl(\lVert h\rVert_{\mu}+3\bigr).$$

Taking $t=s$ and $t=-s$ and dividing by $s>0$ gives $|G-\lambda(\beta\ell_{0}+P)|\le\varepsilon(\lVert h\rVert_{\mu}+3)$. As $\varepsilon>0$ was arbitrary (given $\varepsilon''>0$, take $\varepsilon=\varepsilon''(\lVert h\rVert_{\mu}+3)^{-1}$), Comparison of Real Numbers with Arbitrary Positive Slack §vanishing gives $G=\lambda(\beta\ell_{0}+P)$, and by (1.2),

$$G=\lambda\beta\,L^{a,V}_{\mu}(g).$$

Step 6 (Conclusion). By Steps 5 and 2, with $R=(\lambda\beta)^{-1}\lVert\nabla\chi(\mu)\rVert_{\mu}$, a nonnegative real number not depending on $n$ or $g$,

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

This holds for every $n\in\mathbb{N}$ and every $g\in C^{2}_{b}(\mathbb{R}^{n})$. Since $\mu\in\mathcal{P}_{2}(X)$ and $V$ is integrable with respect to $\mu$, A Bound on the Gibbs Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Fisher Information Relative to the Gibbs Measure §fisher shows that $\mu$ has a relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$. As $\mu\in\mathcal{D}$, $\mu\in\mathcal{D}_{\Sigma}$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain. Since $\chi$, $\lambda$ and the local maximiser $\mu$ were arbitrary, the pair has regular penalised maxima (Noise Penalty Pairs with Regular Penalised Maxima §regular). $\blacksquare$