The Lift of the Second Moment: a Quadratic Function of Class $C^2$ with Gradient 2X and Translation Laplacian 2d

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 $\mathcal{P}_{2}(\mathbb{R}^{d})$ the Wasserstein space; the class $C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$, the gradient $DU(X)$ and the Hessian $D^{2}U(X)$ of a function $U$ on $L^{2}(\Omega;\mathbb{R}^{d})$ are as fixed in The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §space, and $I$ is the identity form on $L^{2}(\Omega;\mathbb{R}^{d})$. 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 $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be the second moment, $u(\mu)=M_{2}(\mu)$, a real number for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, and let $U$ be its lift. Then the following hold.

1. (The lift)¶ $U(X)=\lVert X\rVert_{L^{2}}^{2}$ for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$.

2. (Gradient and Hessian)¶ $U\in C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$, with $DU(X)=2X$ and $D^{2}U(X)=2I$ for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$. In particular $u$ is continuously $L$-differentiable, with $L$-gradient $2X$ at $X$.

3. (The translation Laplacian)¶ $U$ is twice continuously differentiable along translations, and $\Delta_{\mathrm{tr}}U(X)=2d$ for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$, the natural number $d$ read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background §numbers.