Proof of Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian

Each result cited is universally quantified over the data in its own statement, and is used here for the real Hilbert space $E=L^{2}(\Omega;\mathbb{R}^{m})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i), which is open in itself, so that Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space and The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space apply to $\Phi$ on it. Throughout, $X_{a}=Z+c_{a}$ for $a\in\mathbb{R}^{m}$, so that $\phi_{Z}(a)=\Phi(X_{a})$. The set $\mathbb{R}^{m}$ is open by claim 1 of Euclidean Space Is Open in Itself, so Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ and Functions of Class Cᵏ on an Open Subset of ℝⁿ apply to functions on it; continuity in the sense of Continuity at a Point for Maps Between Euclidean Spaces required there is continuity for the Euclidean distance, which by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions is the notion used below.

Claim 1. For $a,b\in\mathbb{R}^{m}$ and $t\in\mathbb{R}$ one has

$$c_{a}+c_{b}=c_{a+b},\qquad t\,c_{a}=c_{ta},\qquad c_{0_{\mathbb{R}^{m}}}=\mathbf{0},$$

where $\mathbf 0$ is the zero vector of $E$: the pointwise sum of the constant maps with values $a$ and $b$ is the constant map with value $a+b$, and likewise for scalar multiples and for the value $0_{\mathbb{R}^{m}}$, and by The Space of Square-Integrable Random Vectors (ii) the class of a pointwise sum is the sum of the classes and the class of a scalar multiple is the scalar multiple of the class. In particular $X_{a}+t\,c_{e_{i}}=X_{a+te_{i}}$ for $i\in[m]$, where the point $a+te_{i}$ is obtained from $a$ by replacing its $i$th component by $a_{i}+t$, by Sum of Points of $\mathbb{R}^n$ and Euclidean Space and Real Matrices: Standing Notation (vi), and $X_{0_{\mathbb{R}^{m}}}=Z$. By The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (iii) and Euclidean Space and Real Matrices: Standing Notation (vi),

$$\lVert c_{a}\rVert_{L^{2}}=\lVert a\rVert,\qquad\lVert c_{e_{i}}\rVert_{L^{2}}=\lVert e_{i}\rVert=1\qquad(i\in[m]).$$

Consequently the map $J=J_{Z}:\mathbb{R}^{m}\to E$, $J(a)=X_{a}$, satisfies $d_{L^{2}}(J(a),J(b))=\lVert c_{a}-c_{b}\rVert_{L^{2}}=\lVert c_{a-b}\rVert_{L^{2}}=\lVert a-b\rVert=d_{E}(a,b)$, the distances being those of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (iii) and Euclidean Space and Real Matrices: Standing Notation (i); so $J$ is Lipschitz with constant $1$ and hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous. This proves claim 1.

Claim 2. Let $a\in\mathbb{R}^{m}$. By The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (iii), read with $m$ in place of $d$, the laws of $Z+c_{a}$ and of $Z'+c_{a}$ are the push-forwards of $\mathcal{L}(Z)$ and of $\mathcal{L}(Z')$ by the translation $\tau_{a}$, hence are equal. Since $\Phi$ is law-invariant, Law-Invariant Function on the Space of Square-Integrable Random Vectors (i) gives $\Phi(Z'+c_{a})=\Phi(Z+c_{a})$, that is $\phi_{Z'}(a)=\phi_{Z}(a)$. As $a$ was arbitrary, $\phi_{Z'}=\phi_{Z}$; being the same function, it is of class $C^{k}$ on $\mathbb{R}^{m}$ under either name exactly when it is under the other, with the same partial derivatives of every order at every point. This proves claim 2.

Claim 3. We first record a division step. Let $A,L,h\in\mathbb{R}$ with $h\ne0$ and let $\varepsilon$ be positive with $|A-hL|\le\tfrac{\varepsilon}{2}|h|$. Since $|h|\,|h^{-1}|=|hh^{-1}|=|1|=1$ by claim 4 of Properties of the Absolute Value in an Ordered Field and Absolute Value in an Ordered Field, the number $|h^{-1}|$ is the multiplicative inverse of $|h|$, which is nonzero by claim 1 of Properties of the Absolute Value in an Ordered Field; hence, by claim 4 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field applied with the nonnegative multiplier $|h^{-1}|$,

$$\Bigl|\frac{A}{h}-L\Bigr|=\bigl|(A-hL)h^{-1}\bigr|=|A-hL|\,|h^{-1}|\le\tfrac{\varepsilon}{2}<\varepsilon ,$$

the last inequality by claim 8 of Elementary Order Arithmetic in an Ordered Field.

Now assume $\Phi\in C^{1}(E)$ and fix $a\in\mathbb{R}^{m}$ and $i\in[m]$, and put $L=\langle D\Phi(X_{a}),c_{e_{i}}\rangle_{L^{2}}$. Let $\varepsilon\in\mathbb{R}$ be positive. Since $\Phi$ is differentiable at $X_{a}$, Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (i) applied with $\tfrac{\varepsilon}{2}$, positive by claim 8 of Elementary Order Arithmetic in an Ordered Field, provides a positive $\delta$ such that every $w\in E$ with $\lVert w\rVert_{L^{2}}<\delta$ satisfies

$$\bigl|\Phi(X_{a}+w)-\Phi(X_{a})-\langle D\Phi(X_{a}),w\rangle_{L^{2}}\bigr|\le\tfrac{\varepsilon}{2}\lVert w\rVert_{L^{2}} .$$

Let $h\in\mathbb{R}$ satisfy $0<|h|<\delta$ and take $w=h\,c_{e_{i}}$, so that $\lVert w\rVert_{L^{2}}=|h|\,\lVert c_{e_{i}}\rVert_{L^{2}}=|h|$ by Elementary Identities in a Real Inner Product Space (iv), and $X_{a}+w=X_{a+he_{i}}$ by claim 1. Since $\langle D\Phi(X_{a}),h\,c_{e_{i}}\rangle_{L^{2}}=hL$ by Elementary Identities in a Real Inner Product Space (i), the display reads

$$\bigl|\phi_{Z}(a+he_{i})-\phi_{Z}(a)-hL\bigr|\le\tfrac{\varepsilon}{2}|h| ,$$

and the division step gives $\bigl|\tfrac{1}{h}(\phi_{Z}(a+he_{i})-\phi_{Z}(a))-L\bigr|<\varepsilon$. By Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ this says exactly that $\partial_{i}\phi_{Z}(a)$ exists and equals $L$, the point $a+he_{i}$ being the one obtained from $a$ by replacing its $i$th component by $a_{i}+h$.

It remains to see that $\phi_{Z}$ is of class $C^{1}$ on $\mathbb{R}^{m}$. For $i\in[m]$ let $\Psi_{i}:E\to\mathbb{R}$ be the function $\Psi_{i}(X)=\langle D\Phi(X),c_{e_{i}}\rangle_{L^{2}}$, so that $\phi_{Z}=\Phi\circ J$ and $\partial_{i}\phi_{Z}=\Psi_{i}\circ J$ by the previous paragraph. The function $\Phi$ is continuous on $E$ by claim 2 of Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space, its gradient map $D\Phi:E\to E$ is continuous by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space (i), and the map $X\mapsto\langle X,c_{e_{i}}\rangle_{L^{2}}$ is Lipschitz with constant $\lVert c_{e_{i}}\rVert_{L^{2}}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (vii), hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous. Claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, applied to the continuous map $D\Phi$ and to the continuous function $X\mapsto\langle X,c_{e_{i}}\rangle_{L^{2}}$, gives that $\Psi_{i}$ is continuous on $E$; applying that claim again, to the continuous map $J$ of claim 1 and to $\Psi_{i}$, and once more to $J$ and to $\Phi$, the functions $\partial_{i}\phi_{Z}=\Psi_{i}\circ J$, for each $i\in[m]$, and $\phi_{Z}=\Phi\circ J$ are continuous at every point of $\mathbb{R}^{m}$. Hence $\phi_{Z}$ is continuous with continuous partial derivatives of every index, so is of class $C^{1}$ on $\mathbb{R}^{m}$ by clauses 1 and 3 of Functions of Class Cᵏ on an Open Subset of ℝⁿ. This proves claim 3.

Claim 4. By The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors (i), read with $m$ in place of $d$, the hypothesis says that $\phi_{Z}$ is of class $C^{2}$ on $\mathbb{R}^{m}$, so its Hessian matrix $D^{2}\phi_{Z}(0_{\mathbb{R}^{m}})$ is defined and lies in $\mathcal{S}(m)$ by Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii), its entry in row $i$ and column $l$ being $\partial_{i}\partial_{l}\phi_{Z}(0_{\mathbb{R}^{m}})$ by The Hessian Matrix of a Function of Class C². By The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors (ii) the translation Laplacian of $\Phi$ at $Z$ is $\Delta_{\mathrm{tr}}\Phi(Z)=\Delta\phi_{Z}(0_{\mathbb{R}^{m}})$, which by The Laplacian of a Twice Continuously Differentiable Function (i) is the sum of the $m$ numbers $\partial_{i}\partial_{i}\phi_{Z}(0_{\mathbb{R}^{m}})$, that is of the diagonal entries of $D^{2}\phi_{Z}(0_{\mathbb{R}^{m}})$; by Trace of a Real Square Matrix this sum is $\mathrm{tr}\,D^{2}\phi_{Z}(0_{\mathbb{R}^{m}})$. This proves claim 4.