Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, assume that the probability space $(\Omega,\mathcal{F},P)$ is rich. The Wasserstein space $\mathcal{P}_{2}(\mathbb{R}^{d})$ and the real Hilbert space $L^{2}(\Omega;\mathbb{R}^{d})$ with its differential calculus are as fixed there, the law map is $\Lambda$, the constant classes $c_{a}$ and translations $\tau_{a}$ are those of that clause, and for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the space $L^{2}(\mu;\mathbb{R}^{d})$ of square-integrable vector fields is the one fixed there, with the tangent space $T_{\mu}\subseteq L^{2}(\mu;\mathbb{R}^{d})$ at $\mu$. For $\varphi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, its lift is the function $\Phi=\varphi\circ\Lambda$ on $L^{2}(\Omega;\mathbb{R}^{d})$; for $\eta\in L^{2}(\mu;\mathbb{R}^{d})$ and $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$, the class $\eta\circ X\in L^{2}(\Omega;\mathbb{R}^{d})$ is the composition of that clause. Being twice continuously differentiable along translations at a point, and on the whole space, are as defined there; $\mathcal{S}(d)$ and the Hessian matrix $D^{2}\phi(a)$ of a function $\phi$ of class $C^{2}$ on $\mathbb{R}^{d}$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i) and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii), the symbol $D$ being read on $L^{2}(\Omega;\mathbb{R}^{d})$ and on $\mathbb{R}^{d}$ as Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation prescribes; and Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian is read with $m=d$. The test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ of that setting and their gradient maps $\nabla\psi$ keep their notation; the test functions and the symbol $\nabla\varphi(\mu)$ introduced below take a function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ as argument, which determines which is meant.

1. (Test function)(i) A function $\varphi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ is a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ if it has the following four properties, the first three of them properties of its lift $\Phi$.

(a) $\Phi$ belongs to the class $C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$; that is, $\varphi$ is continuously $L$-differentiable.

(b) With $D\Phi$ the gradient map of (a): for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ there is an $\eta\in T_{\mu}$ such that $D\Phi(X)=\eta\circ X$ for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$.

(c) $\Phi$ is twice continuously differentiable along translations at every point of $L^{2}(\Omega;\mathbb{R}^{d})$.

(d) $\varphi$ is continuous.

2. (The intrinsic gradient)(ii) Let $\varphi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Then exactly one $\eta$ has the property in clause 1(b). Indeed, one exists by that property, and by The Wasserstein Distance and the Mean-Square Distance of Random Vectors (iii) there is an $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$; if $\eta$ and $\eta'$ both have the property, then $\eta\circ X=D\Phi(X)=\eta'\circ X$, so $(\eta-\eta')\circ X$ is the zero class by the linearity of the composition in Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i), whence $\lVert\eta-\eta'\rVert_{\mu}=\lVert(\eta-\eta')\circ X\rVert_{L^{2}}=0$ by the norm preservation in that clause, and $\eta=\eta'$ by Elementary Identities in a Real Inner Product Space (iii). This unique $\eta$ is written $\nabla\varphi(\mu)$ and called the intrinsic gradient of $\varphi$ at $\mu$; it is an element of $T_{\mu}$, hence of $L^{2}(\mu;\mathbb{R}^{d})$.

3. (The translation Hessian)(iii) Let $\varphi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $X\in L^{2}(\Omega;\mathbb{R}^{d})$ satisfy $\mathcal{L}(X)=\mu$, which exists by The Wasserstein Distance and the Mean-Square Distance of Random Vectors (iii), and let $\phi_{X}:\mathbb{R}^{d}\to\mathbb{R}$ be the function $\phi_{X}(a)=\Phi(X+c_{a})$, whose value at $a$ is $\varphi\bigl((\tau_{a})_{\#}\mu\bigr)$ because $\mathcal{L}(X+c_{a})=(\tau_{a})_{\#}\mu$ by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (vi). By clause 1(c) and Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (iv) the Hessian matrix $D^{2}\phi_{X}(0_{\mathbb{R}^{d}})$ is defined and belongs to $\mathcal{S}(d)$; and since $\Phi$ is law-invariant by Basic Properties of the Lift: Law Invariance, the Correspondence on a Rich Space, and Transfer of Boundedness, Lipschitz Constants and Continuity (i), it does not depend on the choice of $X$, by Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (ii). The translation Hessian of $\varphi$ at $\mu$ is

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