The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, so that the reference measure is $\rho=\gamma_{c}$, let $V$ be an admissible cylindrical potential with noise gradient $\nabla_{a}V$, let $\beta\in\mathbb{R}$ be positive, and let $\gamma^{V}_{\beta}$ be the Gibbs measure of $V$ at temperature $\beta$. Finite relative entropy and $H(\cdot\,|\,\cdot)$ are those of Relative Entropy of Probability Measures §relative-entropy; the relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$ are those of The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential and The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential; $Z^{a}_{\mu}$ is the noise score field and $T^{a}_{\mu}$ the noise tangent space.

1. (Hypothesis)¶ There is a positive $\kappa\in\mathbb{R}$ with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$.

2. (The penalty domain)¶ $\mathcal{D}$ is the set of the $\mu\in\mathcal{P}(X)$ that have finite relative entropy with respect to $\gamma^{V}_{\beta}$. Each such $\mu$ has finite relative entropy with respect to $\gamma_{c}$ by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §entropy, so, under the hypothesis of clause 1, $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property §inclusion.

3. (The score domain)¶ $\mathcal{D}_{\Sigma}$ is the set of the $\mu\in\mathcal{D}$ that have a relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$, these notions being applicable by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §domain. For $\mu\in\mathcal{D}_{\Sigma}$, $V$ 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 and $\mu\in\mathcal{P}_{2}(X)$ by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §domain, so by 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 $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$. Hence $Z^{a}_{\mu}\in T^{a}_{\mu}$ by The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §tangent, the class of $\nabla_{a}V$ belongs to $T^{a}_{\mu}$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §tangent, and, $T^{a}_{\mu}$ being a linear subspace of $L^{2}(\mu;X^{a})$ by Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace §subspace, $\beta\,Z^{a}_{\mu}+\nabla_{a}V\in T^{a}_{\mu}$.

4. (The Gibbs entropy pair)¶ Under the hypothesis of clause 1, the Gibbs entropy pair with potential $V$ and temperature $\beta$ is the quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ consisting of the sets $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ of clauses 2 and 3 and the functions

$$\mathcal{E}:\mathcal{D}\to\mathbb{R},\quad\mathcal{E}(\mu)=\beta\,H(\mu\,|\,\gamma^{V}_{\beta}),\qquad\qquad\Sigma(\mu)=\beta\,Z^{a}_{\mu}+\nabla_{a}V\in T^{a}_{\mu}\quad(\mu\in\mathcal{D}_{\Sigma}).$$