The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors

In the settings of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, let $L^{2}(\Omega;\mathbb{R}^{d})$ be the space of classes of square-integrable random vectors and $c_{a}$, for $a\in\mathbb{R}^{d}$, the constant random vector with value $a$. The class $C^{2}$ on a Euclidean open set and the origin $0_{\mathbb{R}^{d}}$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii) and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i), and the Laplacian $\Delta\phi$ of a function $\phi$ of class $C^{2}$ on $\mathbb{R}^{d}$ is that of The Laplacian of a Twice Continuously Differentiable Function (i); the set $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space Is Open in Itself. The letter $\mu$ denotes a probability measure; a scalar written $\mu$ in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation is written $t$ here.

Let $\Phi:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$ and $X\in L^{2}(\Omega;\mathbb{R}^{d})$, and let $\phi_{X}:\mathbb{R}^{d}\to\mathbb{R}$ be the function

$$\phi_{X}(a)=\Phi(X+c_{a}).$$

1. (Twice continuous differentiability along translations)(i) The function $\Phi$ is twice continuously differentiable along translations at $X$ if $\phi_{X}$ is of class $C^{2}$ on $\mathbb{R}^{d}$.

2. (The translation Laplacian)(ii) If $\Phi$ is twice continuously differentiable along translations at $X$, the translation Laplacian of $\Phi$ at $X$ is the real number

$$\Delta_{\mathrm{tr}}\Phi(X)=\Delta\phi_{X}(0_{\mathbb{R}^{d}}).$$

3. (On the whole space)(iii) The function $\Phi$ is twice continuously differentiable along translations if it is so at every $X\in L^{2}(\Omega;\mathbb{R}^{d})$; its translation Laplacian map is then the function $\Delta_{\mathrm{tr}}\Phi:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$, $X\mapsto\Delta_{\mathrm{tr}}\Phi(X)$.