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

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $m\in\mathbb{N}$ satisfy $1\le m$, let $\Phi:L^{2}(\Omega;\mathbb{R}^{m})\to\mathbb{R}$ and let $Z\in L^{2}(\Omega;\mathbb{R}^{m})$. The real Hilbert space $L^{2}(\Omega;\mathbb{R}^{m})$ is open in itself by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i), so that the differential calculus fixed there applies to $\Phi$ on it: that $\Phi$ is differentiable at a point $X$, its gradient $D\Phi(X)\in L^{2}(\Omega;\mathbb{R}^{m})$ there, and the class $C^{1}(L^{2}(\Omega;\mathbb{R}^{m}))$ is as defined there. Let $c_{a}$ be the constant class with value $a\in\mathbb{R}^{m}$, and let

$$\phi_{Z}:\mathbb{R}^{m}\to\mathbb{R},\qquad\phi_{Z}(a)=\Phi(Z+c_{a}).$$

Partial derivatives, the classes $C^{1}$ and $C^{2}$ on $\mathbb{R}^{m}$ and the Hessian matrix $D^{2}\phi_{Z}(a)\in\mathcal{S}(m)$ of a function of class $C^{2}$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii), the Laplacian $\Delta\phi_{Z}$ is that of The Laplacian of a Twice Continuously Differentiable Function (i), the standard basis vectors $e_{i}$ of $\mathbb{R}^{m}$, the initial segment $[m]$, the origin $0_{\mathbb{R}^{m}}$ and the trace $\mathrm{tr}$ are as in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (v), and the Euclidean norm $\lVert\cdot\rVert$ and Euclidean distance $d_{E}$ are as fixed there. Then the following hold.

1. (Constant classes and translation)(i) For all $a,b\in\mathbb{R}^{m}$ and $t\in\mathbb{R}$,

$$c_{a}+c_{b}=c_{a+b},\qquad t\,c_{a}=c_{ta},\qquad\lVert c_{a}\rVert_{L^{2}}=\lVert a\rVert,$$

and $c_{0_{\mathbb{R}^{m}}}$ is the zero vector of $L^{2}(\Omega;\mathbb{R}^{m})$. Consequently the map $J_{Z}:\mathbb{R}^{m}\to L^{2}(\Omega;\mathbb{R}^{m})$ with $J_{Z}(a)=Z+c_{a}$ satisfies $J_{Z}(0_{\mathbb{R}^{m}})=Z$ and $d_{L^{2}}(J_{Z}(a),J_{Z}(b))=d_{E}(a,b)$, and is continuous.

2. (Law invariance)(ii) Suppose $\Phi$ is law-invariant, that definition being read with $m$ in place of the dimension $d$ named there, and let $Z'\in L^{2}(\Omega;\mathbb{R}^{m})$ satisfy $\mathcal{L}(Z')=\mathcal{L}(Z)$. Then $\phi_{Z'}=\phi_{Z}$. In particular, if $\phi_{Z}$ is of class $C^{2}$ on $\mathbb{R}^{m}$ then so is $\phi_{Z'}$, with the same Hessian matrix at every point.

3. (The translation derivative)(iii) Suppose $\Phi\in C^{1}(L^{2}(\Omega;\mathbb{R}^{m}))$. Then $\phi_{Z}$ is of class $C^{1}$ on $\mathbb{R}^{m}$, with

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

4. (The translation Hessian and its trace)(iv) Suppose $\Phi$ is twice continuously differentiable along translations at $Z$, that definition being read with $m$ in place of the dimension $d$ named there; that is, $\phi_{Z}$ is of class $C^{2}$ on $\mathbb{R}^{m}$. Then $D^{2}\phi_{Z}(0_{\mathbb{R}^{m}})\in\mathcal{S}(m)$, and the translation Laplacian of $\Phi$ at $Z$ is

$$\Delta_{\mathrm{tr}}\Phi(Z)=\mathrm{tr}\,D^{2}\phi_{Z}(0_{\mathbb{R}^{m}}).$$