Lifted Test Functions: Algebra, Quadratics, and the Lifts of Test Functions on the Wasserstein Space

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, lifted test functions on $L^{2}(\Omega;\mathbb{R}^{d})$, their gradients $D\Phi(X)$ and their translation Hessians $H_{\Phi}(X)$ are those of that definition. The space $L^{2}(\Omega;\mathbb{R}^{d})$ with its inner product $\langle\cdot,\cdot\rangle_{L^{2}}$ and norm $\lVert\cdot\rVert_{L^{2}}$, the constant classes $c_{a}$ and the law $\mathcal{L}(X)$ of a class are those of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i); the identity matrix $I_{d}$ and the zero matrix $0_{d}$ of $\mathcal{S}(d)$ are those of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iii); and $\Lambda$ is the law map. Sums and real multiples of real-valued functions on $L^{2}(\Omega;\mathbb{R}^{d})$ are formed pointwise. Then the following hold.

1. (Sums and real multiples)(i) Let $\Phi$ and $\Psi$ be lifted test functions and let $t\in\mathbb{R}$. Then $\Phi+\Psi$ and $t\,\Phi$ are lifted test functions, and every $X\in L^{2}(\Omega;\mathbb{R}^{d})$ satisfies

$$D(\Phi+\Psi)(X)=D\Phi(X)+D\Psi(X),\qquad H_{\Phi+\Psi}(X)=H_{\Phi}(X)+H_{\Psi}(X),$$ $$D(t\,\Phi)(X)=t\,D\Phi(X),\qquad H_{t\,\Phi}(X)=t\,H_{\Phi}(X).$$

2. (Affine functions)(ii) Let $V\in L^{2}(\Omega;\mathbb{R}^{d})$ and $c\in\mathbb{R}$, and let $\Phi:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$ be given by

$$\Phi(X)=\langle V,X\rangle_{L^{2}}+c .$$

Then $\Phi$ is a lifted test function, and $D\Phi(X)=V$ and $H_{\Phi}(X)=0_{d}$ for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$.

3. (Multiples of a squared distance)(iii) Let $\alpha\in\mathbb{R}$, let $Z\in L^{2}(\Omega;\mathbb{R}^{d})$, and let $\Phi:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$ be given by

$$\Phi(X)=\tfrac{\alpha}{2}\,\lVert X-Z\rVert_{L^{2}}^{2}.$$

Then $\Phi$ is a lifted test function, and $D\Phi(X)=\alpha\,(X-Z)$ and $H_{\Phi}(X)=\alpha\,I_{d}$ for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$.

4. (Lifts of test functions on the Wasserstein space)(iv) Assume that $(\Omega,\mathcal{F},P)$ is rich and let $\varphi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, with intrinsic gradient $\nabla\varphi(\mu)$ and translation Hessian $H_{\varphi}(\mu)$ at $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Then $\varphi\circ\Lambda$ is a lifted test function, and every $X\in L^{2}(\Omega;\mathbb{R}^{d})$ satisfies

$$D(\varphi\circ\Lambda)(X)=\nabla\varphi(\mathcal{L}(X))\circ X,\qquad H_{\varphi\circ\Lambda}(X)=H_{\varphi}(\mathcal{L}(X)),$$

the composition $\eta\circ X$ of a vector field $\eta$ over $\mathcal{L}(X)$ with $X$ being that of Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i).