Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mathcal{P}^{a}_{\rho}$ be the set of The Measures Noise-Connected to the Reference Measure §space, which contains $\rho$ 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, and let $W_{a}$ be the noise Wasserstein distance, a metric on $\mathcal{P}^{a}_{\rho}$ 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 §metric; in particular $W_{a}(\mu,\rho)$ is a nonnegative real number for every $\mu\in\mathcal{P}^{a}_{\rho}$. For $\mu\in\mathcal{P}^{a}_{\rho}$, $L^{2}(\mu;X^{a})$ is the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, with inner product $\langle\cdot,\cdot\rangle_{\mu}$, and $T^{a}_{\mu}\subseteq L^{2}(\mu;X^{a})$ is the noise tangent space at $\mu$. $\mathcal{F}C^{1}_{b}(X)$ is the set of bounded $C^{1}$ cylindrical functions, and for $\psi\in\mathcal{F}C^{1}_{b}(X)$, $\nabla_{a}\psi:X\to X^{a}$ is its noise gradient, a measurable map that is square-integrable with respect to every $\mu\in\mathcal{P}(X)$ by that clause; its class in $L^{2}(\mu;X^{a})$ is again written $\nabla_{a}\psi$. $\mathcal{F}C^{2}_{b}(X)$ is the set of bounded $C^{2}$ cylindrical functions, a subset of $\mathcal{F}C^{1}_{b}(X)$: for every $n\in\mathbb{N}$ a function of class $C^{2}$ on $\mathbb{R}^{n}$ is of class $C^{1}$ there by clause 2 of C^k Maps on a Euclidean Open Set, so that $C^{2}_{b}(\mathbb{R}^{n})\subseteq C^{1}_{b}(\mathbb{R}^{n})$ by Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded and Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, and the inclusion follows from the two definitions of cylindrical functions. For $\mu\in\mathcal{P}^{a}_{\rho}$, $\psi\in\mathcal{F}C^{1}_{b}(X)$ and $t\in\mathbb{R}$, $\mathrm{id}+t\,\nabla_{a}\psi:X\to X$ denotes the map $x\mapsto x+t\,\nabla_{a}\psi(x)$; it is Borel and $(\mathrm{id}+t\,\nabla_{a}\psi)_{\#}\mu\in\mathcal{P}^{a}_{\rho}$ by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §displacement and Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §displacement-connected, applied with $h$ the class of $\nabla_{a}\psi$ and its representative $\nabla_{a}\psi$. Open intervals $(p,q)$ are those of that definition, and differentiability of a real function on an interval at an interior point, with its derivative, is that of Derivative at an Interior Point.

(Noise penalty pair)¶ A noise penalty pair on $\mathcal{P}^{a}_{\rho}$ is a quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ consisting of two sets

$$\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho},$$

called the penalty domain $\mathcal{D}$ and the score domain $\mathcal{D}_{\Sigma}$, a function $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, called the penalty, and a function $\Sigma$ assigning to each $\mu\in\mathcal{D}_{\Sigma}$ an element $\Sigma(\mu)\in T^{a}_{\mu}$, called the score, such that the following four conditions hold.

1. (Nonempty score domain)¶ $\mathcal{D}_{\Sigma}$ is nonempty.

2. (Lower bound by the distance to the reference measure)¶ There is a nonnegative $C\in\mathbb{R}$ such that

$$-C\bigl(1+W_{a}(\mu,\rho)^{2}\bigr)\le\mathcal{E}(\mu)\qquad\text{for every }\mu\in\mathcal{D}.$$

3. (The score is the first variation of the penalty)¶ For every $\mu\in\mathcal{D}_{\Sigma}$ and every $\psi\in\mathcal{F}C^{2}_{b}(X)$ there is a real number $t_{0}>0$ such that $(\mathrm{id}+t\,\nabla_{a}\psi)_{\#}\mu\in\mathcal{D}$ for every $t$ in the open interval $(-t_{0},t_{0})$, and the function

$$(-t_{0},t_{0})\to\mathbb{R},\qquad t\mapsto\mathcal{E}\bigl((\mathrm{id}+t\,\nabla_{a}\psi)_{\#}\mu\bigr),$$

is differentiable at $0$ with derivative $\langle\Sigma(\mu),\nabla_{a}\psi\rangle_{\mu}$. Here $(-t_{0},t_{0})$ is an interval and $0$ is an interior point of it by Basic Facts about Intervals of the Real Line and Their Interior Points §open-interval, since $t_{0}>0$ gives $-t_{0}<-0=0<t_{0}$ by Elementary Order Arithmetic in an Ordered Field §sign-reversal; and the derivative is unique by Uniqueness of the Derivative at an Interior Point, an interval being order-convex, the defining conditions of the two notions being the same, applied with $u=-(t_{0}\cdot2^{-1})$ and $v=t_{0}\cdot2^{-1}$, which lie in $(-t_{0},t_{0})$ and satisfy $u<0<v$ because $0<t_{0}\cdot2^{-1}<t_{0}$ by Elementary Order Arithmetic in an Ordered Field §halving and by Elementary Order Arithmetic in an Ordered Field §sign-reversal.

4. (Density of the score domain)¶ For every $\mu\in\mathcal{D}$ and every positive $\varepsilon\in\mathbb{R}$ there is $\nu\in\mathcal{D}_{\Sigma}$ with $W_{a}(\nu,\mu)<\varepsilon$.