Proof of First Variations of a Test Function on the Wasserstein Space Along Gradient Displacements and Along Translations

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

By The Wasserstein Distance and the Mean-Square Distance of Random Vectors §onto there is an $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$; fix one, and let $\Phi=\varphi\circ\Lambda$ be the lift of $\varphi$.

Claim 1. Let $a\in\mathbb{R}^{d}$. By The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants, $\mathcal{L}(X+c_{a})=(\tau_{a})_{\#}\mathcal{L}(X)=(\tau_{a})_{\#}\mu$ and $\lVert c_{a}\rVert_{L^{2}}=\lVert a\rVert$. Since $X+c_{a}\in L^{2}(\Omega;\mathbb{R}^{d})$, its law belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations §law, so $(\tau_{a})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Moreover $(X+c_{a})-X=(X+c_{a})+(-X)=c_{a}+\bigl(X+(-X)\bigr)=c_{a}$, by claim 2 of Elementary Identities in a Vector Space for the notation and the commutativity, associativity, inverse and identity axioms of the vector space $L^{2}(\Omega;\mathbb{R}^{d})$, so The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §law-map gives

Claim 2. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and let $t_{0}\in\mathbb{R}$ be positive; write $G_{t}=\mathrm{id}+t\,\nabla\psi$ and $g(t)=\varphi\bigl((G_{t})_{\#}\mu\bigr)$ for $t\in(-t_{0},t_{0})$. Let $Z=\nabla\psi\circ X\in L^{2}(\Omega;\mathbb{R}^{d})$ be the composition of the class of $\nabla\psi$ in $L^{2}(\mu;\mathbb{R}^{d})$ with $X$; by that clause $\lVert Z\rVert_{L^{2}}=\lVert\nabla\psi\rVert_{\mu}$.

For $t\in\mathbb{R}$ the map $G_{t}$ is Borel by The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound §borel, so $\mathcal{L}(G_{t}\circ X)=(G_{t})_{\#}\mathcal{L}(X)=(G_{t})_{\#}\mu$ by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition. A representative of $G_{t}\circ X$ takes the value $X(\omega)+t\,\nabla\psi(X(\omega))$ at $\omega\in\Omega$, and, the sum and the real multiple in $L^{2}(\Omega;\mathbb{R}^{d})$ being formed pointwise on representatives by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations, so does a representative of $X+tZ$; the two classes are therefore equal, $G_{t}\circ X=X+tZ$. Hence

By condition (a) of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian §test and The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1, $\Phi$ is differentiable at $X$ with gradient $D\Phi(X)$, and $D\Phi(X)=\nabla\varphi(\mu)\circ X$ by condition (b) and Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian §gradient. Writing $L=\langle\nabla\varphi(\mu),\nabla\psi\rangle_{\mu}$, the preservation of inner products in Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity §composition gives

Let $\varepsilon\in\mathbb{R}$ be positive and put $c=\lVert Z\rVert_{L^{2}}$, so $0\le c$ and $0<c+1$ by claim 3 of Elementary Order Arithmetic in an Ordered Field applied to $0<1$ (claim 6 of that lemma) and $0\le c$. The number $\varepsilon(c+1)^{-1}$ is positive by claims 7 and 5 of that lemma, so Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable provides a positive $\rho\in\mathbb{R}$ with

Let $\eta$ be the lesser of $t_{0}$ and $\rho(c+1)^{-1}$ (claim 9 of Elementary Order Arithmetic in an Ordered Field), positive because it is one of them, and let $h\in\mathbb{R}$ satisfy $0<|h|<\eta$ and $h\in(-t_{0},t_{0})$. Then, by Elementary Identities in a Real Inner Product Space §homogeneity and claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative multiplier $|h|$, and by claim 10 of Elementary Order Arithmetic in an Ordered Field with the positive multiplier $c+1$,

Taking $V=hZ$ and using $\langle D\Phi(X),hZ\rangle_{L^{2}}=h\,L$ (Elementary Identities in a Real Inner Product Space §bilinear) therefore gives

Now $\frac{g(h)-g(0)}{h}-L=\bigl(g(h)-g(0)-h\,L\bigr)h^{-1}$, and $|h|\,|h^{-1}|=|h\,h^{-1}|=|1|=1$ by claim 4 of Properties of the Absolute Value in an Ordered Field and Absolute Value in an Ordered Field, so multiplying the last display by $|h|^{-1}$, nonnegative by claim 7 of Elementary Order Arithmetic in an Ordered Field, yields by claim 5 of Elementary Arithmetic in an Ordered Field

the final inequality because $c<c+1$ gives $c\,(c+1)^{-1}<1$ and hence $\varepsilon(c+1)^{-1}c<\varepsilon$, by claim 10 of Elementary Order Arithmetic in an Ordered Field with the positive multipliers $(c+1)^{-1}$ and $\varepsilon$. Since $\varepsilon$ was an arbitrary positive real number, $g$ is differentiable at $0$ with $g'(0)=L$ by Derivative at an Interior Point, $0$ lying in $(-t_{0},t_{0})$ because $-t_{0}<-0=0<t_{0}$ by claim 4 of Elementary Order Arithmetic in an Ordered Field and being interior to it by Basic Facts about Intervals of the Real Line and Their Interior Points §open-interval.

Claim 3. The function $a\mapsto\varphi\bigl((\tau_{a})_{\#}\mu\bigr)$ is the function $\phi_{X}$ with $\phi_{X}(a)=\Phi(X+c_{a})$ attached to $\Phi$ at $X$ in The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors §translations, because $\mathcal{L}(X+c_{a})=(\tau_{a})_{\#}\mu$ by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants and $\Phi=\varphi\circ\Lambda$. By condition (c) of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian §test it is of class $C^{2}$ on $\mathbb{R}^{d}$, and its Hessian matrix at $0_{\mathbb{R}^{d}}$ is $H_{\varphi}(\mu)$ by Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian §hessian, which also shows that this matrix does not depend on the choice of $X$.