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 (ii), the set $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space Is Open in Itself, and the Laplacian $\Delta f$ is that of The Laplacian of a Twice Continuously Differentiable Function (i). 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

$$|\partial_{i}f(x)|\le M_{1}\qquad\text{and}\qquad|\partial_{j}\partial_{i}f(x)|\le M_{2}\qquad\text{for all }x\in\mathbb{R}^{d}\text{ and all }i,j\in[d].$$

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 Functions of Class Cᵏ on an Open Subset of ℝⁿ, 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 (ii); $\Delta f$ is Borel by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, 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 (ii); 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 Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (ii). Then the following hold.

1. (Integrability and the lift)(i) 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

$$U(X)=\mathbb{E}[f\circ X].$$

2. (The $L$-gradient)(ii) 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

$$DU(X)=Df\circ X\qquad\text{for every }X\in L^{2}(\Omega;\mathbb{R}^{d}).$$

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

$$\lVert DU(X)-DU(Y)\rVert_{L^{2}}\le d\,M_{2}\,\lVert X-Y\rVert_{L^{2}},$$

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

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

$$\Delta_{\mathrm{tr}}U(X)=\mathbb{E}[\Delta f\circ X]=\int_{\mathbb{R}^{d}}\Delta f\,d\mathcal{L}(X).$$