The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian

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 $C^{2}$, the partial derivatives $\partial_{i}f$, the iterated partial derivatives $\partial_{j}\partial_{i}f$ and the gradient $Df(x)\in\mathbb{R}^{d}$ of a function $f$ on $\mathbb{R}^{d}$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, the set $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and the Laplacian $\Delta f$ is that of The Laplacian of a Twice Continuously Differentiable Function §laplacian. 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 $f:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{d}$, and let $M_{1},M_{2}\in\mathbb{R}$ be nonnegative with

The functions $f$, $\partial_{i}f$ and $\partial_{j}\partial_{i}f$ are continuous at every point of $\mathbb{R}^{d}$ by clauses 1 and 2 of C^k Maps on a Euclidean Open Set, hence Borel by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps; $\Delta f$ is Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and $Df:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is Borel by the componentwise criterion of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps; so for every random vector $X$ in $\mathbb{R}^{d}$ the maps $f\circ X$ and $\Delta f\circ X$ are random variables and $Df\circ X$ is a random vector, by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition. Then the following hold.

1. (Integrability and the lift)¶ For every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the function $f$ is integrable with respect to $\mu$. Hence $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, $u(\mu)=\int_{\mathbb{R}^{d}}f\,d\mu$, is defined, and its lift $U$ satisfies, for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$ and every representative of $X$, that $f\circ X$ is integrable and

2. (The $L$-gradient)¶ For every $X\in L^{2}(\Omega;\mathbb{R}^{d})$ and every representative of $X$, the random vector $Df\circ X$ is square-integrable, and its class, again written $Df\circ X$, is the same for all representatives of $X$. The function $u$ is $L$-differentiable, with $L$-gradient

3. (The gradient map is Lipschitz)¶ For all $X,Y\in L^{2}(\Omega;\mathbb{R}^{d})$,

the natural number $d$ read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background §numbers; in particular $u$ is continuously $L$-differentiable.

4. (The translation Laplacian)¶ $U$ is twice continuously differentiable along translations, and for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$ and every representative of $X$,