Sums, Real Multiples and Differences of Test Functions on the Wasserstein Space

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, assume that $(\Omega,\mathcal{F},P)$ is rich. Test functions on $\mathcal{P}_{2}(\mathbb{R}^{d})$, their intrinsic gradients $\nabla\varphi(\mu)$, elements of the tangent space $T_{\mu}$, and their translation Hessians $H_{\varphi}(\mu)$, elements of the set $\mathcal{S}(d)$ of symmetric real $d\times d$ matrices, are those of that definition. Sums and real multiples of matrices are those of Euclidean Space and Real Matrices: Standing Notation (ii), and sums and real multiples in $L^{2}(\mu;\mathbb{R}^{d})$ those of the real Hilbert space fixed there. In this statement test function always means a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\nabla\varphi(\mu)$ its intrinsic gradient; the test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (iii) and their gradient maps $\nabla\psi$ are not used. For $\varphi,\chi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ and $c\in\mathbb{R}$, the functions $\varphi+\chi$, $c\varphi$ and $\varphi-\chi$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$ take the values $\varphi(\mu)+\chi(\mu)$, $c\,\varphi(\mu)$ and $\varphi(\mu)-\chi(\mu)$ at $\mu$, and $-\varphi$ is the function $(-1)\varphi$.

Let $\varphi$ and $\chi$ be test functions and let $c\in\mathbb{R}$. Then the following hold.

1. (Sums and real multiples)(i) The functions $\varphi+\chi$ and $c\varphi$ are test functions, and for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$

$$\nabla(\varphi+\chi)(\mu)=\nabla\varphi(\mu)+\nabla\chi(\mu),\qquad H_{\varphi+\chi}(\mu)=H_{\varphi}(\mu)+H_{\chi}(\mu),$$ $$\nabla(c\varphi)(\mu)=c\,\nabla\varphi(\mu),\qquad H_{c\varphi}(\mu)=c\,H_{\varphi}(\mu).$$

2. (Differences)(ii) The functions $\varphi-\chi$ and $-\varphi$ are test functions, and for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$

$$\nabla(\varphi-\chi)(\mu)=\nabla\varphi(\mu)-\nabla\chi(\mu),\qquad H_{\varphi-\chi}(\mu)=H_{\varphi}(\mu)-H_{\chi}(\mu),$$ $$\nabla(-\varphi)(\mu)=-\nabla\varphi(\mu),\qquad H_{-\varphi}(\mu)=-H_{\varphi}(\mu).$$