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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ be the Wasserstein space of the fixed dimension $d$ with its distance, with the second moment $M_{2}$ and the translations $\tau_{a}$, and for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ let $T_{\mu}$ be the tangent space at $\mu$, a subset of the space $L^{2}(\mu;\mathbb{R}^{d})$ of square-integrable vector fields. Push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv). Being of class $C^{2}$ on $\mathbb{R}^{d}$ and the Hessian matrix $D^{2}\phi(a)$ of such a function $\phi$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii), $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space Is Open in Itself; $0_{\mathbb{R}^{d}}$ is the origin of $\mathbb{R}^{d}$, and $\mathcal{S}(d)$ is the set of symmetric real $d\times d$ matrices. Test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and their gradient maps $\nabla\psi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ are as fixed there, the class of $\nabla\psi$ in $L^{2}(\mu;\mathbb{R}^{d})$ being again written $\nabla\psi$, with the inner product $\langle\cdot,\cdot\rangle_{\mu}$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii). For every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and every $t\in\mathbb{R}$, $\mathrm{id}+t\,\nabla\psi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ denotes the map $x\mapsto x+t\,\nabla\psi(x)$ formed from the gradient map, written $G_{t}$ in The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound; it is Borel and its push-forward $(\mathrm{id}+t\,\nabla\psi)_{\#}\mu$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound (i). Open intervals $(p,q)$ are those of that definition, and differentiability of a real function on an interval at an interior point is that of Derivative of a Real Function on an Interval. The letter $\Sigma$ is used here for the score map of a penalty pair and is not bound elsewhere in this setting. The score $\Sigma(\mu)$ of a penalty pair is a datum of the pair and is not the score $\xi_{\mu}$ of a measure of finite Fisher information, which keeps its own notation.

(Penalty pair)(i) A penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ is a quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ consisting of two sets

$$\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}_{2}(\mathbb{R}^{d}),$$

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 $\Sigma(\mu)\in T_{\mu}$, called the score, such that the following five conditions hold.

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

2. (Lower bound by the second moment)(iii) There is a nonnegative $C\in\mathbb{R}$ such that

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

3. (Twice differentiable along translations)(iv) For every $\mu\in\mathcal{D}$ and every $a\in\mathbb{R}^{d}$ the push-forward $(\tau_{a})_{\#}\mu$ belongs to $\mathcal{D}$, and the function

$$e_{\mu}:\mathbb{R}^{d}\to\mathbb{R},\qquad e_{\mu}(a)=\mathcal{E}\bigl((\tau_{a})_{\#}\mu\bigr),$$

is of class $C^{2}$ on $\mathbb{R}^{d}$.

4. (The score is the first variation of the penalty)(v) For every $\mu\in\mathcal{D}_{\Sigma}$ and every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ there is a real number $t_{0}>0$ such that $(\mathrm{id}+t\,\nabla\psi)_{\#}\mu\in\mathcal{D}$ for every $t$ in the open interval $(-t_{0},t_{0})$, and the function

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

is differentiable at $0$ with derivative $\langle\Sigma(\mu),\nabla\psi\rangle_{\mu}$; here $0$ is an interior point of the interval $(-t_{0},t_{0})$ by Basic Facts about Intervals of the Real Line (iv), since $-t_{0}<-0=0<t_{0}$ (claim 4 of Elementary Order Arithmetic in an Ordered Field), and the derivative is unique by Uniqueness of the Derivative of a Real Function on an Interval, an interval being order-convex, the defining conditions of the two notions being the same.

5. (Density of the score domain)(vi) For every $\mu\in\mathcal{D}$ and every positive $\varepsilon\in\mathbb{R}$ there is $\nu\in\mathcal{D}_{\Sigma}$ with $W_{2}(\nu,\mu)<\varepsilon$.

(The translation Hessian of the penalty)(vii) Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair and let $\mu\in\mathcal{D}$. By condition 3 the Hessian matrix $D^{2}e_{\mu}(0_{\mathbb{R}^{d}})$ is defined, and it is symmetric by claim 2 of Equality of Mixed Partial Derivatives, and Symmetry and Continuity of the Hessian. It is called the translation Hessian of the penalty at $\mu$ and written

$$H_{\mathcal{E}}(\mu)=D^{2}e_{\mu}(0_{\mathbb{R}^{d}})\in\mathcal{S}(d).$$