Proof of Lifted Test Functions: Algebra, Quadratics, and the Lifts of Test Functions on the Wasserstein Space

Each result cited is universally quantified over the data in its own statement and is applied here to the data named in the statement of the lemma. Throughout, $E=L^{2}(\Omega;\mathbb{R}^{d})$ is a real inner product space, indeed a real Hilbert space, and is open in itself, by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i); the differential calculus of Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus (iv) and the results Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space, Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space and Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ are therefore in force for $E$ with $U=E$. For a function $\Phi$ on $E$ and $X\in E$ we write $\phi_{X}(a)=\Phi(X+c_{a})$, as in Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians (ii). The class $C^{2}$ on $\mathbb{R}^{d}$ is that of Functions of Class Cᵏ on an Open Subset of ℝⁿ, so that a function on $\mathbb{R}^{d}$ is of class $C^{2}$ exactly when its partial derivatives of orders one and two exist at every point and are continuous, and the Hessian matrix has the second partial derivatives as its entries by The Hessian Matrix of a Function of Class C². Every member of $C^{2}(E)$ belongs to $C^{1}(E)$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space (ii). Finally Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian is read with $m=d$.

A preliminary identity. For $a,b\in\mathbb{R}^{d}$,

$$\langle c_{a},c_{b}\rangle_{L^{2}}=a\cdot b .$$

Indeed, $c_{a}+c_{b}=c_{a+b}$ by Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (i), and $\lVert c_{w}\rVert_{L^{2}}=\lVert w\rVert$ for every $w\in\mathbb{R}^{d}$ by that same claim. Expanding $\lVert c_{a}+c_{b}\rVert_{L^{2}}^{2}$ by Elementary Identities in a Real Inner Product Space (v) gives $\lVert c_{a}\rVert_{L^{2}}^{2}+2\langle c_{a},c_{b}\rangle_{L^{2}}+\lVert c_{b}\rVert_{L^{2}}^{2}$, while $\lVert c_{a+b}\rVert_{L^{2}}^{2}=\lVert a+b\rVert^{2}=\lVert a\rVert^{2}+2\,a\cdot b+\lVert b\rVert^{2}$ by the same identity read in $\mathbb{R}^{d}$, whose inner product is the dot product and whose norm is the Euclidean norm, related by claim 1 of that lemma. Subtracting the equal terms $\lVert a\rVert^{2}$ and $\lVert b\rVert^{2}$ from the two expressions and dividing by the positive $2$ gives the identity.

Claim 1. Let $\Phi,\Psi$ be lifted test functions and let $t\in\mathbb{R}$.

By Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians (i)(a) both belong to $C^{1}(E)$, so $\Phi+\Psi\in C^{1}(E)$ with $D(\Phi+\Psi)(X)=D\Phi(X)+D\Psi(X)$ by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (ii), and $t\Phi\in C^{1}(E)$ with $D(t\Phi)(X)=t\,D\Phi(X)$ by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (iii). This is property (a) for both functions, and the two gradient identities.

For property (b), let $X\in E$. Writing $\chi$ for $\Phi+\Psi$, one has $\chi_{X}(a)=\Phi(X+c_{a})+\Psi(X+c_{a})=\phi_{X}(a)+\psi_{X}(a)$ for every $a\in\mathbb{R}^{d}$, where $\phi_{X}$ and $\psi_{X}$ are of class $C^{2}$ on $\mathbb{R}^{d}$ by Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians (i)(b) and The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors (i). By claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions the sum $\chi_{X}$ is of class $C^{2}$ on $\mathbb{R}^{d}$, and by claim 1 of that lemma, applied first to $\phi_{X}$ and $\psi_{X}$ and then to their partial derivatives, each partial derivative of $\chi_{X}$ of order at most two is the sum of the corresponding partial derivatives of $\phi_{X}$ and $\psi_{X}$; so $\chi_{X}$ is of class $C^{2}$, which is property (b) for $\Phi+\Psi$ by The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors (iii), and, the entries of a Hessian matrix being the second partial derivatives, $D^{2}\chi_{X}(0_{\mathbb{R}^{d}})=D^{2}\phi_{X}(0_{\mathbb{R}^{d}})+D^{2}\psi_{X}(0_{\mathbb{R}^{d}})$, that is $H_{\Phi+\Psi}(X)=H_{\Phi}(X)+H_{\Psi}(X)$. The same argument with claims 3 and 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, applied to the scalar multiple of $\phi_{X}$ by $t$, gives property (b) for $t\Phi$ and $H_{t\Phi}(X)=t\,H_{\Phi}(X)$.

Claim 2. By Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ (i), read with $p=V$, the function $\Phi$ belongs to $C^{2}(E)$, hence to $C^{1}(E)$, with $D\Phi(X)=V$ for every $X\in E$; this is property (a) and the gradient identity.

Let $X\in E$. Since $\Phi\in C^{1}(E)$, Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (iii) gives that $\phi_{X}$ is of class $C^{1}$ on $\mathbb{R}^{d}$ with

$$\partial_{i}\phi_{X}(a)=\langle D\Phi(X+c_{a}),c_{e_{i}}\rangle_{L^{2}}=\langle V,c_{e_{i}}\rangle_{L^{2}}\qquad(a\in\mathbb{R}^{d},\ i\in[d]),$$

a value not depending on $a$. Each $\partial_{i}\phi_{X}$ is therefore a constant function on $\mathbb{R}^{d}$, which is of class $C^{2}$ by claim 2 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, and all of whose partial derivatives are $0$, directly from Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$, every difference quotient of a constant function vanishing. Hence all partial derivatives of $\phi_{X}$ of orders one and two exist and are continuous, so $\phi_{X}$ is of class $C^{2}$ on $\mathbb{R}^{d}$, which is property (b) by The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors (i) and The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors (iii), and every entry of $D^{2}\phi_{X}(0_{\mathbb{R}^{d}})$ is $0$, that is $H_{\Phi}(X)=0_{d}$.

Claim 3. By Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ (iii), read with $y_{0}=Z$, the function $\Phi$ belongs to $C^{2}(E)$, hence to $C^{1}(E)$, with $D\Phi(X)=\alpha\,(X-Z)$ for every $X\in E$; this is property (a) and the gradient identity.

Let $X\in E$. By Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (iii) the function $\phi_{X}$ is of class $C^{1}$ on $\mathbb{R}^{d}$ with, for $a\in\mathbb{R}^{d}$ and $i\in[d]$,

$$\partial_{i}\phi_{X}(a)=\langle D\Phi(X+c_{a}),c_{e_{i}}\rangle_{L^{2}}=\langle\alpha\,(X-Z)+\alpha\,c_{a},c_{e_{i}}\rangle_{L^{2}}=\alpha\,\langle X-Z,c_{e_{i}}\rangle_{L^{2}}+\alpha\,(a\cdot e_{i}),$$

the second equality because $X+c_{a}-Z=(X-Z)+c_{a}$ and the third by the bilinearity of the inner product (Elementary Identities in a Real Inner Product Space (i)) together with the preliminary identity. Now $a\cdot e_{i}=a_{i}$, the $i$-th coordinate of $a$, by Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$ and Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iii). So $\partial_{i}\phi_{X}$ is the sum of the constant $\alpha\langle X-Z,c_{e_{i}}\rangle_{L^{2}}$ and the function $a\mapsto\alpha\,a_{i}$; by claims 2 and 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions it is of class $C^{2}$ on $\mathbb{R}^{d}$, with

$$\partial_{j}\partial_{i}\phi_{X}(a)=\alpha\,\partial_{j}(a\mapsto a_{i})=\begin{cases}\alpha,&j=i,\\ 0,&j\ne i,\end{cases}$$

the partial derivatives of a constant function being $0$ and those of the coordinate function $a\mapsto a_{i}$ being $1$ in the direction $i$ and $0$ otherwise, in each case directly from Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$. These are constant, hence continuous, so $\phi_{X}$ is of class $C^{2}$ on $\mathbb{R}^{d}$, which is property (b), and by The Hessian Matrix of a Function of Class C² the matrix $D^{2}\phi_{X}(0_{\mathbb{R}^{d}})$ has $\alpha$ in each diagonal entry and $0$ elsewhere, that is $H_{\Phi}(X)=\alpha\,I_{d}$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iii).

Claim 4. Let $\varphi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\Phi=\varphi\circ\Lambda$ be its lift, the function used in Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (i). Property (a) of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (i) says that $\Phi\in C^{1}(E)$, which is property (a) of Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians (i); property (c) there says that $\Phi$ is twice continuously differentiable along translations at every point of $E$, which by The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors (iii) is property (b). Hence $\varphi\circ\Lambda$ is a lifted test function.

Let $X\in E$ and $\mu=\mathcal{L}(X)$. By property (b) of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (i) there is $\eta\in T_{\mu}$ with $D\Phi(X')=\eta\circ X'$ for every $X'\in E$ with $\mathcal{L}(X')=\mu$, and by Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (ii) this $\eta$ is unique and is $\nabla\varphi(\mu)$. Taking $X'=X$ gives $D(\varphi\circ\Lambda)(X)=\nabla\varphi(\mathcal{L}(X))\circ X$, the composition being that of Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i).

For the Hessians, Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (iii) defines $H_{\varphi}(\mu)$ as $D^{2}\phi_{X}(0_{\mathbb{R}^{d}})$ for the function $\phi_{X}(a)=\Phi(X+c_{a})$ and any $X$ with $\mathcal{L}(X)=\mu$, and Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians (ii) defines $H_{\varphi\circ\Lambda}(X)$ as $D^{2}\phi_{X}(0_{\mathbb{R}^{d}})$ for the same function $\phi_{X}$. The two are therefore the same matrix, which is the asserted identity.