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 (iii) 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 (vi), $\mathcal{L}(X+c_{a})=(\tau_{a})_{\#}\mathcal{L}(X)=(\tau_{a})_{\#}\mu$ 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 (ii), 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 (v) gives

$$W_{2}\bigl((\tau_{a})_{\#}\mu,\mu\bigr)=W_{2}\bigl(\mathcal{L}(X+c_{a}),\mathcal{L}(X)\bigr)\le\lVert (X+c_{a})-X\rVert_{L^{2}}=\lVert c_{a}\rVert_{L^{2}}=\lVert a\rVert .$$

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 (i), so $\mathcal{L}(G_{t}\circ X)=(G_{t})_{\#}\mathcal{L}(X)=(G_{t})_{\#}\mu$ by Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (ii). 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 (ii), so does a representative of $X+tZ$; the two classes are therefore equal, $G_{t}\circ X=X+tZ$. Hence

$$g(t)=\Phi(X+tZ)\qquad\bigl(t\in(-t_{0},t_{0})\bigr).$$

By condition (a) of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (i) and The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space (i), $\Phi$ 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 (ii). 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 (i) gives

$$\langle D\Phi(X),Z\rangle_{L^{2}}=\langle\nabla\varphi(\mu)\circ X,\nabla\psi\circ X\rangle_{L^{2}}=L .$$

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 (i) provides a positive $\rho\in\mathbb{R}$ with

$$\bigl|\Phi(X+V)-\Phi(X)-\langle D\Phi(X),V\rangle_{L^{2}}\bigr|\le\varepsilon(c+1)^{-1}\lVert V\rVert_{L^{2}}\qquad\text{for every }V\in L^{2}(\Omega;\mathbb{R}^{d})\text{ with }\lVert V\rVert_{L^{2}}<\rho .$$

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 (iv) 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 $c+1$,

$$\lVert hZ\rVert_{L^{2}}=|h|\,c\le|h|\,(c+1)<\eta\,(c+1)\le\rho .$$

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

$$\bigl|g(h)-g(0)-h\,L\bigr|\le\varepsilon(c+1)^{-1}|h|\,c .$$

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

$$\left|\frac{g(h)-g(0)}{h}-L\right|\le\varepsilon(c+1)^{-1}c<\varepsilon ,$$

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 of a Real Function on an Interval, $0$ 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 (iv).

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 (i), because $\mathcal{L}(X+c_{a})=(\tau_{a})_{\#}\mu$ by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (vi) and $\Phi=\varphi\circ\Lambda$. By condition (c) of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (i) 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 (iii), which also shows that this matrix does not depend on the choice of $X$.