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

Each result cited is universally quantified over the data in its own statement.

Let $\Phi=\varphi\circ\Lambda$ and $\Xi=\chi\circ\Lambda$ be the lifts of $\varphi$ and $\chi$, and let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

Claim 1. Since $\Lambda(Y)=\mathcal{L}(Y)$ for $Y\in L^{2}(\Omega;\mathbb{R}^{d})$, the lift of $\varphi+\chi$ has the value $\varphi(\Lambda(Y))+\chi(\Lambda(Y))=\Phi(Y)+\Xi(Y)$ at $Y$, and the lift of $c\varphi$ the value $c\,\Phi(Y)$; these lifts are therefore $\Phi+\Xi$ and $c\Phi$. We verify the four conditions of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (i) for $\varphi+\chi$ and for $c\varphi$.

(a) $\Phi$ and $\Xi$ belong to $C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$ by condition (a) for $\varphi$ and for $\chi$, the set $L^{2}(\Omega;\mathbb{R}^{d})$ being open in itself by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (iii). Hence $\Phi+\Xi$ and $c\Phi$ belong to $C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$, with

$$D(\Phi+\Xi)(Y)=D\Phi(Y)+D\Xi(Y),\qquad D(c\Phi)(Y)=c\,D\Phi(Y)\qquad\bigl(Y\in L^{2}(\Omega;\mathbb{R}^{d})\bigr),$$

by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (ii) and Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (iii).

(b) Let $Y\in L^{2}(\Omega;\mathbb{R}^{d})$ satisfy $\mathcal{L}(Y)=\mu$. By condition (b) for $\varphi$ and for $\chi$ and by Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (ii), $D\Phi(Y)=\nabla\varphi(\mu)\circ Y$ and $D\Xi(Y)=\nabla\chi(\mu)\circ Y$. The map $\zeta\mapsto\zeta\circ Y$ is linear by Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i), so by (a)

$$D(\Phi+\Xi)(Y)=\bigl(\nabla\varphi(\mu)+\nabla\chi(\mu)\bigr)\circ Y,\qquad D(c\Phi)(Y)=\bigl(c\,\nabla\varphi(\mu)\bigr)\circ Y .$$

The fields $\nabla\varphi(\mu)$ and $\nabla\chi(\mu)$ lie in $T_{\mu}$, a linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (i), so $\nabla\varphi(\mu)+\nabla\chi(\mu)$ and $c\,\nabla\varphi(\mu)$ lie in $T_{\mu}$ as well. Thus condition (b) holds for $\varphi+\chi$ and for $c\varphi$ at $\mu$, with these fields, and $\mu$ was an arbitrary element of $\mathcal{P}_{2}(\mathbb{R}^{d})$.

(c) Let $Y\in L^{2}(\Omega;\mathbb{R}^{d})$ and let $\phi_{Y},\xi_{Y}:\mathbb{R}^{d}\to\mathbb{R}$ be the functions $\phi_{Y}(a)=\Phi(Y+c_{a})$ and $\xi_{Y}(a)=\Xi(Y+c_{a})$ attached to $\Phi$ and to $\Xi$ at $Y$ in The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors (i); by condition (c) for $\varphi$ and for $\chi$ they are of class $C^{2}$ on $\mathbb{R}^{d}$. The functions attached in the same way to $\Phi+\Xi$ and to $c\Phi$ are $\phi_{Y}+\xi_{Y}$ and $c\,\phi_{Y}$, of class $C^{2}$ on $\mathbb{R}^{d}$ by claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions. So condition (c) holds for $\varphi+\chi$ and for $c\varphi$ at every point of $L^{2}(\Omega;\mathbb{R}^{d})$.

(d) $\varphi$ and $\chi$ are continuous by condition (d), hence so are $\varphi+\chi$ and $c\varphi$ by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space.

Therefore $\varphi+\chi$ and $c\varphi$ are test functions, and by the uniqueness of the intrinsic gradient in Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (ii) the fields exhibited in (b) are the intrinsic gradients:

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

For the translation Hessians, fix $Y$ with $\mathcal{L}(Y)=\mu$, which exists by The Wasserstein Distance and the Mean-Square Distance of Random Vectors (iii), and keep the notation of (c). The partial derivatives of $\phi_{Y}$ and of $\xi_{Y}$ exist at every point of $\mathbb{R}^{d}$, so by claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, $\partial_{j}(\phi_{Y}+\xi_{Y})=\partial_{j}\phi_{Y}+\partial_{j}\xi_{Y}$ and $\partial_{j}(c\,\phi_{Y})=c\,\partial_{j}\phi_{Y}$ as functions on $\mathbb{R}^{d}$, for each index $j$; applying that claim once more, at $0_{\mathbb{R}^{d}}$, to these functions gives

$$\partial_{i}\partial_{j}(\phi_{Y}+\xi_{Y})(0_{\mathbb{R}^{d}})=\partial_{i}\partial_{j}\phi_{Y}(0_{\mathbb{R}^{d}})+\partial_{i}\partial_{j}\xi_{Y}(0_{\mathbb{R}^{d}}),\qquad\partial_{i}\partial_{j}(c\,\phi_{Y})(0_{\mathbb{R}^{d}})=c\,\partial_{i}\partial_{j}\phi_{Y}(0_{\mathbb{R}^{d}})$$

for all indices $i$ and $j$. The entries of a Hessian matrix are exactly these second partial derivatives, and matrices with the same entries are equal, so by Sum of Real Matrices, Scalar Multiple of a Real Matrix and Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (iii),

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

Claim 2. In $\mathbb{R}$ one has $(-1)x=-(1x)=-x$ by claim 2 of Zero Products and Elementary Identities in a Field and the multiplicative identity axiom of the field $\mathbb{R}$, and $x+(-1)y=x+(-y)=x-y$ by the notation for the difference fixed in Zero Products and Elementary Identities in a Field, so $-\varphi=(-1)\varphi$ and $\varphi-\chi=\varphi+(-1)\chi$ as functions on $\mathcal{P}_{2}(\mathbb{R}^{d})$. By claim 1, $(-1)\chi$ is a test function with intrinsic gradient $(-1)\nabla\chi(\mu)$ and translation Hessian $(-1)H_{\chi}(\mu)$ at $\mu$, and then $\varphi+(-1)\chi$ and $(-1)\varphi$ are test functions with

$$\nabla(\varphi-\chi)(\mu)=\nabla\varphi(\mu)+(-1)\nabla\chi(\mu),\qquad H_{\varphi-\chi}(\mu)=H_{\varphi}(\mu)+(-1)H_{\chi}(\mu),$$ $$\nabla(-\varphi)(\mu)=(-1)\nabla\varphi(\mu),\qquad H_{-\varphi}(\mu)=(-1)H_{\varphi}(\mu).$$

In the real vector space $L^{2}(\mu;\mathbb{R}^{d})$ one has $(-1)\zeta=-\zeta$ by claim 5 of Elementary Identities in a Vector Space and $\zeta+(-1)\omega=\zeta+(-\omega)=\zeta-\omega$ by claim 2 of that lemma; for matrices, $(-1)B$ has the entries $-B_{ij}$ by Scalar Multiple of a Real Matrix and claim 2 of Zero Products and Elementary Identities in a Field, so $(-1)B=-B$ and $A+(-1)B=A-B$ by Sum of Real Matrices and Difference of Real Matrices. Rewriting the four displayed identities accordingly gives the assertion.