Proof of The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions

Each result cited is universally quantified over the data in its own statement. Throughout, $\pi_{i}:\mathbb{R}^{d}\to\mathbb{R}$, $\pi_{i}(x)=x_{i}$, is the $i$th coordinate function, smooth by claim 2 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, hence continuous (claim 3 of Euclidean Space Is Open in Itself) and Borel (claim 3 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets); $e_{i}$ is the $i$th standard basis vector, so $x\cdot e_{i}=x_{i}$ (as recorded there) and $\lVert e_{i}\rVert=1$ (claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, the coordinates of $e_{i}$ being $0$ except for one $1$); sums over $i\in[d]$ are the finite sums of Properties of Finite Sums, and "integrable" is integrable with respect to the measure named. The integral of a finite sum of integrable functions is the sum of the integrals, by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion applied along the recursion of claim 1 of Properties of Finite Sums; we refer to this as finite additivity. For a probability measure the integral of a constant is that constant (Simple Functions, Nonnegative Simple Functions, and the Simple Integral), and $\lVert x\rVert^{2}=\sum_{i}x_{i}^{2}$ for $x\in\mathbb{R}^{d}$ (claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square). Two facts from Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) are used: $|x\cdot y|\le\tfrac12(\lVert x\rVert^{2}+\lVert y\rVert^{2})$, and the Borel measurability of $x\mapsto\lVert x\rVert^{2}$; with $y=e_{i}$ the first gives

$$|x_{i}|\le\tfrac12\bigl(\lVert x\rVert^{2}+1\bigr)\qquad(x\in\mathbb{R}^{d}).\tag{B}$$

Proof of claim 1.

Integrability and the mean. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By (B), $|\pi_{i}|\le\tfrac12(\lVert\cdot\rVert^{2}+1)$, and the right side is Borel with $\int\tfrac12(\lVert x\rVert^{2}+1)\mu(dx)=\tfrac12(M_{2}(\mu)+1)<\infty$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (i); hence $\int|\pi_{i}|\,d\mu<\infty$ by the monotonicity in that claim, and $\pi_{i}$ is integrable (Integrable Functions and Their Lebesgue Integral). So $m(\mu)$ is defined.

Lift. Let $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$ and fix a representative; its law is $\mu$ by The Space of Square-Integrable Random Vectors (iv). The coordinate $X_{i}$ is $\pi_{i}\circ X$ (Random Vector and Its Law (ii)), and Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iii) gives that $\pi_{i}\circ X$ is integrable with $\mathbb{E}[X_{i}]=\int\pi_{i}\,d\mu=m(\mu)_{i}$.

Bound. By Hölder's Inequality (i) with $p=q=2$ (conjugate exponents in the sense of Conjugate Exponents and Young's Inequality (i), as $\tfrac12+\tfrac12=1$), applied to $\pi_{i}$ and to the constant function $1$ on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (i)), $\int|\pi_{i}|\,d\mu\le\bigl(\int\pi_{i}^{2}\,d\mu\bigr)^{1/2}\bigl(\int1\,d\mu\bigr)^{1/2}=\bigl(\int\pi_{i}^{2}\,d\mu\bigr)^{1/2}$, where $\pi_{i}^{2}\le\lVert\cdot\rVert^{2}$ is integrable. With $|m(\mu)_{i}|=|\int\pi_{i}\,d\mu|\le\int|\pi_{i}|\,d\mu$ (claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion) and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $m(\mu)_{i}^{2}\le\int\pi_{i}^{2}\,d\mu$; summing over $i$ (claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers) and using finite additivity, $\lVert m(\mu)\rVert^{2}\le\int\sum_{i}\pi_{i}^{2}\,d\mu=\int\lVert x\rVert^{2}\mu(dx)=M_{2}(\mu)$.

Translation. Let $a\in\mathbb{R}^{d}$ and $\mu_{a}=(\tau_{a})_{\#}\mu$, an element of $\mathcal{P}_{2}(\mathbb{R}^{d})$ by Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields (v). By claim 2 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (the image measure of $\mu$ under the Borel map $\tau_{a}$, Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv)), $\int\pi_{i}\,d\mu_{a}=\int\pi_{i}\circ\tau_{a}\,d\mu=\int(x_{i}+a_{i})\,\mu(dx)=m(\mu)_{i}+a_{i}$, the last step by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion; that is, $m(\mu_{a})=m(\mu)+a$ (sums in $\mathbb{R}^{d}$ being coordinatewise, Euclidean Points as Tuples of Real Numbers).

Lipschitz bound. Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$, so $I(\pi)<\infty$ by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (iii). By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i), $\mu$ and $\nu$ are the image measures of $\pi$ under $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ (Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i)), so claim 2 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables gives that $\pi_{i}\circ\mathrm{pr}_{1}$ and $\pi_{i}\circ\mathrm{pr}_{2}$ are integrable with respect to $\pi$ with $\int\pi_{i}\circ\mathrm{pr}_{1}\,d\pi=m(\mu)_{i}$ and $\int\pi_{i}\circ\mathrm{pr}_{2}\,d\pi=m(\nu)_{i}$. Hence, with $D_{i}=\pi_{i}\circ\mathrm{pr}_{1}-\pi_{i}\circ\mathrm{pr}_{2}$, the function $z\mapsto(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z))_{i}$,

$$m(\mu)_{i}-m(\nu)_{i}=\int D_{i}\,d\pi\tag{M}$$

by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion. As in the bound paragraph (Hoelder with the constant $1$ on the probability space $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\pi)$, and $D_{i}^{2}\le\lVert\mathrm{pr}_{1}-\mathrm{pr}_{2}\rVert^{2}$, which is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) with $\pi$-integral $I(\pi)<\infty$, Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (ii)), $(m(\mu)_{i}-m(\nu)_{i})^{2}\le\int D_{i}^{2}\,d\pi$, and summing, $\lVert m(\mu)-m(\nu)\rVert^{2}\le\int\lVert\mathrm{pr}_{1}-\mathrm{pr}_{2}\rVert^{2}\,d\pi=I(\pi)$. Since $\pi\in\Pi(\mu,\nu)$ was arbitrary, $\lVert m(\mu)-m(\nu)\rVert^{2}$ is a lower bound of $\{I(\pi):\pi\in\Pi(\mu,\nu)\}$, hence at most its greatest lower bound $W_{2}(\mu,\nu)^{2}$ (The Quadratic Wasserstein Distance on Euclidean Space (i), Lower Bound and Greatest Lower Bound), and $\lVert m(\mu)-m(\nu)\rVert\le W_{2}(\mu,\nu)$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Proof of claim 2. Write $m=m(\mu)$. By Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields (v) with $a=-m$, $\bar{\mu}=(\tau_{-m})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and by the translation paragraph of claim 1, $m(\bar{\mu})=m+(-m)=0_{\mathbb{R}^{d}}$.

Second moment. By claim 2 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables for the nonnegative Borel function $\lVert\cdot\rVert^{2}$, $M_{2}(\bar{\mu})=\int\lVert x-m\rVert^{2}\,\mu(dx)$. For every $x$, $\lVert x-m\rVert^{2}=\sum_{i}(x_{i}-m_{i})^{2}=\sum_{i}\bigl(x_{i}^{2}-2m_{i}x_{i}+m_{i}^{2}\bigr)$ (claim 5 of Zero Products and Elementary Identities in a Field), and each summand is integrable; by finite additivity, claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and $\int\pi_{i}\,d\mu=m_{i}$,

$$M_{2}(\bar{\mu})=\sum_{i}\Bigl(\int\pi_{i}^{2}\,d\mu-2m_{i}m_{i}+m_{i}^{2}\Bigr)=M_{2}(\mu)-\lVert m\rVert^{2},$$

using claims 2 and 3 of Properties of Finite Sums.

Composition of translations. For $a,b\in\mathbb{R}^{d}$ and every Borel set $B$, claim 1 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables gives $\bigl((\tau_{b})_{\#}(\tau_{a})_{\#}\mu\bigr)(B)=\mu\bigl(\tau_{a}^{-1}(\tau_{b}^{-1}(B))\bigr)=\mu\bigl((\tau_{b}\circ\tau_{a})^{-1}(B)\bigr)=\bigl((\tau_{a+b})_{\#}\mu\bigr)(B)$, since $\tau_{b}(\tau_{a}(x))=x+a+b=\tau_{a+b}(x)$; thus $(\tau_{b})_{\#}(\tau_{a})_{\#}\mu=(\tau_{a+b})_{\#}\mu$, and $(\tau_{0_{\mathbb{R}^{d}}})_{\#}\mu=\mu$ because $\tau_{0_{\mathbb{R}^{d}}}$ is the identity. Hence $(\tau_{m})_{\#}\bar{\mu}=(\tau_{-m+m})_{\#}\mu=\mu$; and for $a\in\mathbb{R}^{d}$, $m((\tau_{a})_{\#}\mu)=m+a$ by claim 1, so $\overline{(\tau_{a})_{\#}\mu}=(\tau_{-m-a})_{\#}(\tau_{a})_{\#}\mu=(\tau_{-m})_{\#}\mu=\bar{\mu}$.

The lift. Let $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$. By Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (i) (read with its dimension parameter equal to $d$), $c_{-m}=(-1)c_{m}$, which is $-c_{m}$ by claim 5 of Elementary Identities in a Vector Space, so $X-c_{m}=X+c_{-m}$ (claim 2 there), and by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (vi), $\mathcal{L}(X-c_{m})=(\tau_{-m})_{\#}\mathcal{L}(X)=\bar{\mu}$.

The centred coupling. Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, write $c=m(\mu)-m(\nu)$, let $\pi\in\Pi(\mu,\nu)$, and let $S=\tau_{-m(\mu)}$, $T=\tau_{-m(\nu)}$, Borel maps. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (v), $\pi'=(\mathrm{pr}_{1},T\circ\mathrm{pr}_{2})_{\#}\pi\in\Pi(\mu,\bar{\nu})$ and $\bar{\pi}=(S\circ\mathrm{pr}_{1},\mathrm{pr}_{2})_{\#}\pi'\in\Pi(\bar{\mu},\bar{\nu})$. Since $\mathrm{pr}_{1}\circ(u,v)=u$ and $\mathrm{pr}_{2}\circ(u,v)=v$ for a pairing (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i)), one has $(S\circ\mathrm{pr}_{1},\mathrm{pr}_{2})\circ(\mathrm{pr}_{1},T\circ\mathrm{pr}_{2})=(S\circ\mathrm{pr}_{1},T\circ\mathrm{pr}_{2})$, so $\bar{\pi}=(S\circ\mathrm{pr}_{1},T\circ\mathrm{pr}_{2})_{\#}\pi$ by the composition rule for image measures (claim 1 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables, exactly as for translations above). By claim 2 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables and Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (ii),

$$I(\bar{\pi})=\int\bigl\lVert S(\mathrm{pr}_{1}(z))-T(\mathrm{pr}_{2}(z))\bigr\rVert^{2}\,\pi(dz)=\int\bigl\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)-c\bigr\rVert^{2}\,\pi(dz),$$

because $S(x)-T(y)=(x-m(\mu))-(y-m(\nu))=x-y-c$. With $w=\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)$, $\lVert w-c\rVert^{2}=\sum_{i}(w_{i}-c_{i})^{2}=\lVert w\rVert^{2}-2\sum_{i}c_{i}w_{i}+\lVert c\rVert^{2}$ (claim 5 of Zero Products and Elementary Identities in a Field, claims 2 and 3 of Properties of Finite Sums), where $w_{i}=D_{i}(z)$ is integrable (Lipschitz paragraph of claim 1) and $\lVert w\rVert^{2}$ has integral $I(\pi)<\infty$. Integrating with claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, finite additivity and (M),

$$I(\bar{\pi})=I(\pi)-2\sum_{i}c_{i}\bigl(m(\mu)_{i}-m(\nu)_{i}\bigr)+\lVert c\rVert^{2}=I(\pi)-2\lVert c\rVert^{2}+\lVert c\rVert^{2}=I(\pi)-\lVert c\rVert^{2}.$$

By The Quadratic Wasserstein Distance on Euclidean Space (i), $W_{2}(\bar{\mu},\bar{\nu})^{2}\le I(\bar{\pi})$, so $W_{2}(\bar{\mu},\bar{\nu})^{2}+\lVert c\rVert^{2}\le I(\pi)$ (claim 3 of Elementary Arithmetic in an Ordered Field). As $\pi\in\Pi(\mu,\nu)$ was arbitrary, the left side is a lower bound of $\{I(\pi):\pi\in\Pi(\mu,\nu)\}$, hence at most $W_{2}(\mu,\nu)^{2}$ (Lower Bound and Greatest Lower Bound); and since $0\le\lVert c\rVert^{2}$ (Nonnegativity of Squares in an Ordered Field), $W_{2}(\bar{\mu},\bar{\nu})^{2}\le W_{2}(\mu,\nu)^{2}$, so $W_{2}(\bar{\mu},\bar{\nu})\le W_{2}(\mu,\nu)$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

The expectation map. For claims 3 and 4 define $\mathsf{m}:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}^{d}$ by $\mathsf{m}(X)=m(\mathcal{L}(X))$, defined since $\mathcal{L}(X)\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii). By claim 1, $\mathsf{m}(X)_{i}=\mathbb{E}[X_{i}]$ for every representative of $X$. The coordinates of $X+H$ and of $tX$ ($t\in\mathbb{R}$) are $X_{i}+H_{i}$ and $tX_{i}$ (operations being pointwise, The Space of Square-Integrable Random Vectors (ii)), so claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives

$$\mathsf{m}(X+H)=\mathsf{m}(X)+\mathsf{m}(H),\qquad\mathsf{m}(tX)=t\,\mathsf{m}(X),\qquad\mathsf{m}(c_{a})=a,$$

the last because $c_{a}$ has coordinates constant equal to $a_{i}$; hence $\mathsf{m}(X+c_{a})=\mathsf{m}(X)+a$. By claim 1 and The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii),

$$\lVert\mathsf{m}(H)\rVert^{2}\le M_{2}(\mathcal{L}(H))=\lVert H\rVert_{L^{2}}^{2},\qquad\text{so}\qquad\lVert\mathsf{m}(H)\rVert\le\lVert H\rVert_{L^{2}}\tag{L}$$

(claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), and in particular $\lVert\mathsf{m}(X)-\mathsf{m}(Y)\rVert=\lVert\mathsf{m}(X-Y)\rVert\le\lVert X-Y\rVert_{L^{2}}$. Moreover, for $p\in\mathbb{R}^{d}$ and $H\in L^{2}(\Omega;\mathbb{R}^{d})$, by The Space of Square-Integrable Random Vectors (iii), $c_{p}\cdot H=\sum_{i}p_{i}H_{i}$ pointwise and finite additivity,

$$\langle c_{p},H\rangle_{L^{2}}=\mathbb{E}\Bigl[\sum_{i}p_{i}H_{i}\Bigr]=\sum_{i}p_{i}\,\mathbb{E}[H_{i}]=p\cdot\mathsf{m}(H).\tag{C}$$

Finally, by Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (i), $c_{a}+c_{b}=c_{a+b}$, $tc_{a}=c_{ta}$ and $\lVert c_{a}\rVert_{L^{2}}=\lVert a\rVert$, so $\lVert c_{a}-c_{b}\rVert_{L^{2}}=\lVert c_{a-b}\rVert_{L^{2}}=\lVert a-b\rVert$.

Translates of $C^{k}$ functions. For $g:\mathbb{R}^{d}\to\mathbb{R}$ and $b\in\mathbb{R}^{d}$ write $g^{b}=g\circ\tau_{b}$, the function $x\mapsto g(x+b)$. For every $x$ and $i$ the difference quotients of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ for $g^{b}$ at $x$ are those of $g$ at $x+b$, because $g^{b}(x+he_{i})=g(x+b+he_{i})$; hence $\partial_{i}g^{b}$ exists at $x$ if and only if $\partial_{i}g$ exists at $x+b$, and then $\partial_{i}(g^{b})=(\partial_{i}g)^{b}$ (the value of a partial derivative being unique: if $L,L'$ both satisfy the definition then $|L-L'|<2\varepsilon$ for every $\varepsilon>0$ by claim 5 of Properties of the Absolute Value in an Ordered Field, so $L=L'$ by Comparison of Real Numbers with Arbitrary Positive Slack). Also $g^{b}$ is continuous whenever $g$ is, by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, $\tau_{b}$ being continuous since $\lVert\tau_{b}(x)-\tau_{b}(x')\rVert=\lVert x-x'\rVert$. By induction on $k$ (Principle of Induction for the Natural Numbers, on the set of $k\in\mathbb{N}$ such that for every $g$ of class $C^{k}$ on $\mathbb{R}^{d}$ and every $b$ the function $g^{b}$ is of class $C^{k}$ on $\mathbb{R}^{d}$ with $\partial_{i}(g^{b})=(\partial_{i}g)^{b}$ for all $i$), using clauses 1 and 2 of Functions of Class Cᵏ on an Open Subset of ℝⁿ: if $g$ is of class $C^{2}$ then so is $g^{b}$, with $\partial_{j}\partial_{i}(g^{b})=(\partial_{j}\partial_{i}g)^{b}$, so that $D^{2}(g^{b})(x)=D^{2}g(x+b)$ (The Hessian Matrix of a Function of Class C², entrywise).

Proof of claim 3. Let $\phi$ be of class $C^{2}$ on $\mathbb{R}^{d}$ and $\varphi(\mu)=\phi(m(\mu))$, with lift $\Phi=\varphi\circ\Lambda$ (The Lift of a Function on the Wasserstein Space to the Space of Square-Integrable Random Vectors (i)), so $\Phi(X)=\phi(\mathsf{m}(X))$. We verify the four properties of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (i).

(a). Let $X\in L^{2}(\Omega;\mathbb{R}^{d})$, $a=\mathsf{m}(X)$ and $p=D\phi(a)$, the point with coordinates $\partial_{i}\phi(a)$ (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii)). Since $\phi$ is of class $C^{1}$ (clause 2 of Functions of Class Cᵏ on an Open Subset of ℝⁿ), it is differentiable at $a$ with derivative matrix the $1\times d$ matrix of the partials $\partial_{i}\phi(a)$ by Continuous Partial Derivatives Imply Differentiability; so by Differentiability at a Point for Functions Between Euclidean Spaces, for every $\varepsilon>0$ there is $\delta>0$ such that $|\phi(a+h)-\phi(a)-p\cdot h|\le\varepsilon\lVert h\rVert$ whenever $0<\lVert h\rVert<\delta$, and trivially also for $h=0_{\mathbb{R}^{d}}$. Let $H\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\lVert H\rVert_{L^{2}}<\delta$ and put $h=\mathsf{m}(H)$, so $\lVert h\rVert<\delta$ by (L). By the expectation-map paragraph and (C),

$$\bigl|\Phi(X+H)-\Phi(X)-\langle c_{p},H\rangle_{L^{2}}\bigr|=\bigl|\phi(a+h)-\phi(a)-p\cdot h\bigr|\le\varepsilon\lVert h\rVert\le\varepsilon\lVert H\rVert_{L^{2}}.$$

Hence $\Phi$ is differentiable at $X$ with gradient $D\Phi(X)=c_{D\phi(\mathsf{m}(X))}$ (Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (i), Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (ii)). The gradient map is continuous: given $X$ and $\varepsilon>0$, the $d$ continuous functions $\partial_{i}\phi$ give $\delta>0$ with $|\partial_{i}\phi(a')-\partial_{i}\phi(a)|\le\varepsilon(2\sqrt{d})^{-1}$ for all $i$ whenever $\lVert a'-a\rVert<\delta$ (the least of $d$ radii, by claim 9 of Elementary Order Arithmetic in an Ordered Field applied repeatedly; continuity at a point as in Continuous Map Between Metric Spaces); for $\lVert Y-X\rVert_{L^{2}}<\delta$ one has $\lVert\mathsf{m}(Y)-\mathsf{m}(X)\rVert<\delta$ by (L), hence $\lVert D\Phi(Y)-D\Phi(X)\rVert_{L^{2}}^{2}=\lVert D\phi(\mathsf{m}(Y))-D\phi(\mathsf{m}(X))\rVert^{2}\le d\cdot\varepsilon^{2}(4d)^{-1}=\varepsilon^{2}/4$ by claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, so $\lVert D\Phi(Y)-D\Phi(X)\rVert_{L^{2}}\le\varepsilon/2<\varepsilon$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field again, and claim 8 of Elementary Order Arithmetic in an Ordered Field). Thus $\Phi\in C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$ (The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space (i)).

(b). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\eta$ be the class of the constant map with value $p=D\phi(m(\mu))$, which lies in $T_{\mu}$ by Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields (ii). For $X$ with $\mathcal{L}(X)=\mu$, $\mathsf{m}(X)=m(\mu)$, and $\eta\circ X$ (Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i)) is the class of the constant map with value $p$, i.e. $c_{p}=D\Phi(X)$.

(c). Let $X\in L^{2}(\Omega;\mathbb{R}^{d})$ and $\phi_{X}(a)=\Phi(X+c_{a})=\phi(\mathsf{m}(X)+a)=\phi^{\mathsf{m}(X)}(a)$. By the translates paragraph, $\phi_{X}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with $D^{2}\phi_{X}(0_{\mathbb{R}^{d}})=D^{2}\phi(\mathsf{m}(X))$; so $\Phi$ is twice continuously differentiable along translations at every $X$ (The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors (i), The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors (iii)).

(d). The map $m:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}^{d}$ is Lipschitz with constant $1$ by claim 1 (Lipschitz Map Between Metric Spaces, with $W_{2}$ and the Euclidean distance $d_{E}(x,y)=\lVert x-y\rVert$, claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square), hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous; $\phi$ is continuous by claim 3 of Euclidean Space Is Open in Itself; so $\varphi=\phi\circ m$ is continuous by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map.

Hence $\varphi$ is a test function; its intrinsic gradient at $\mu$ is the $\eta$ of (b) (Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (ii)), and its translation Hessian at $\mu$ is $D^{2}\phi_{X}(0_{\mathbb{R}^{d}})=D^{2}\phi(m(\mu))$ for any $X$ with $\mathcal{L}(X)=\mu$ (Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (iii)).

Proof of claim 4. Let $\varphi$ be a test function with lift $\Phi$, and define the centring operator $\mathsf{C}:L^{2}(\Omega;\mathbb{R}^{d})\to L^{2}(\Omega;\mathbb{R}^{d})$ by $\mathsf{C}X=X-c_{\mathsf{m}(X)}$. By the lift paragraph of claim 2, $\mathcal{L}(\mathsf{C}X)=\overline{\mathcal{L}(X)}$, so the lift $\Phi^{\circ}$ of $\varphi^{\circ}$ satisfies $\Phi^{\circ}(X)=\varphi(\overline{\mathcal{L}(X)})=\Phi(\mathsf{C}X)$. By the expectation-map paragraph, $\mathsf{C}(X+H)=\mathsf{C}X+\mathsf{C}H$ and, using claim 2 and The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii), $\lVert\mathsf{C}H\rVert_{L^{2}}^{2}=M_{2}(\overline{\mathcal{L}(H)})=M_{2}(\mathcal{L}(H))-\lVert\mathsf{m}(H)\rVert^{2}\le\lVert H\rVert_{L^{2}}^{2}$, so

$$\lVert\mathsf{C}H\rVert_{L^{2}}\le\lVert H\rVert_{L^{2}}\qquad\text{and}\qquad\lVert\mathsf{C}X-\mathsf{C}Y\rVert_{L^{2}}=\lVert\mathsf{C}(X-Y)\rVert_{L^{2}}\le\lVert X-Y\rVert_{L^{2}}.\tag{N}$$

Moreover, by Elementary Identities in a Real Inner Product Space (i) and (C), $\langle\mathsf{C}H,G\rangle_{L^{2}}=\langle H,G\rangle_{L^{2}}-\mathsf{m}(H)\cdot\mathsf{m}(G)$, which is symmetric in $H$ and $G$; hence

$$\langle\mathsf{C}H,G\rangle_{L^{2}}=\langle H,\mathsf{C}G\rangle_{L^{2}}\qquad(H,G\in L^{2}(\Omega;\mathbb{R}^{d})).\tag{S}$$

(a). Let $X\in L^{2}(\Omega;\mathbb{R}^{d})$ and $Y=\mathsf{C}X$. Given $\varepsilon>0$, let $\delta>0$ be provided by the differentiability of $\Phi$ at $Y$ (Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (i), property (a) of the test function $\varphi$). For $\lVert H\rVert_{L^{2}}<\delta$, (N) gives $\lVert\mathsf{C}H\rVert_{L^{2}}<\delta$, so by (S)

$$\bigl|\Phi^{\circ}(X+H)-\Phi^{\circ}(X)-\langle\mathsf{C}D\Phi(Y),H\rangle_{L^{2}}\bigr|=\bigl|\Phi(Y+\mathsf{C}H)-\Phi(Y)-\langle D\Phi(Y),\mathsf{C}H\rangle_{L^{2}}\bigr|\le\varepsilon\lVert\mathsf{C}H\rVert_{L^{2}}\le\varepsilon\lVert H\rVert_{L^{2}}.$$

Thus $\Phi^{\circ}$ is differentiable at $X$ with $D\Phi^{\circ}(X)=\mathsf{C}\bigl(D\Phi(\mathsf{C}X)\bigr)$. The gradient map $X\mapsto\mathsf{C}(D\Phi(\mathsf{C}X))$ is continuous, as the composition of $\mathsf{C}$ (Lipschitz by (N), hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous), the continuous map $D\Phi$, and $\mathsf{C}$ again (claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map). So $\Phi^{\circ}\in C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$.

(b) and the gradient formula. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $m=m(\mu)$, and let $\eta$ be a representative of $\nabla\varphi(\bar{\mu})\in T_{\bar{\mu}}$. By Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields (v) applied to $\bar{\mu}$, $a=m$ and $\eta$, using $(\tau_{m})_{\#}\bar{\mu}=\mu$ (claim 2): the map $\eta^{\ast}:x\mapsto\eta(x-m)$ is Borel, its class in $L^{2}(\mu;\mathbb{R}^{d})$ does not depend on the representative, and it belongs to $T_{\mu}$. Since $\int\lVert\eta^{\ast}\rVert^{2}d\mu<\infty$ and $|\eta^{\ast}_{i}|\le\tfrac12(\lVert\eta^{\ast}\rVert^{2}+1)$ by (B), the coordinates $\eta^{\ast}_{i}$ are integrable with respect to $\mu$ (as in the first paragraph of the proof of claim 1); since $\eta^{\ast}_{i}=\eta_{i}\circ\tau_{-m}$, claim 2 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables for the image measure $\bar{\mu}=(\tau_{-m})_{\#}\mu$ shows that the coordinate $\eta_{i}$ is integrable with respect to $\bar{\mu}$ with $\int\eta_{i}\,d\bar{\mu}=\int\eta^{\ast}_{i}\,d\mu$. Let $b=\int\eta\,d\bar{\mu}$ be the point with these coordinates; if $\eta'$ is another representative then $\bar{\mu}(\{\eta=\eta'\})=1$, so $\int\eta'_{i}\,d\bar{\mu}=\int\eta_{i}\,d\bar{\mu}$ by The Lebesgue Integral and Null Sets (iii), and $b$ does not depend on the representative. Now let $X$ satisfy $\mathcal{L}(X)=\mu$ and $Y=\mathsf{C}X=X-c_{m}$, so $\mathcal{L}(Y)=\bar{\mu}$ and $D\Phi(Y)=\nabla\varphi(\bar{\mu})\circ Y$ by property (b) of $\varphi$. For a representative of $X$, the map $\omega\mapsto\eta(Y(\omega))=\eta(X(\omega)-m)=\eta^{\ast}(X(\omega))$ shows $\nabla\varphi(\bar{\mu})\circ Y=\eta^{\ast}\circ X$ (Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i)). Its coordinates have expectations $\mathbb{E}[\eta^{\ast}_{i}\circ X]=\int\eta^{\ast}_{i}\,d\mu=b_{i}$ (Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iii)), so $\mathsf{m}(\eta^{\ast}\circ X)=b$ and

$$D\Phi^{\circ}(X)=\mathsf{C}(\eta^{\ast}\circ X)=\eta^{\ast}\circ X-c_{b}=(\eta^{\ast}-b)\circ X,$$

by the linearity of composition 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), the constant map with value $b$ composed with $X$ being $c_{b}$. The class of $\eta^{\ast}-b$, i.e. of $x\mapsto\eta(x-m)-\int\eta\,d\bar{\mu}$, lies in $T_{\mu}$: $\eta^{\ast}\in T_{\mu}$, $b\in T_{\mu}$ by Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields (ii), and $T_{\mu}$ is a linear subspace 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). This gives property (b), and by Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (ii) the intrinsic gradient $\nabla\varphi^{\circ}(\mu)$ is this class.

(c). For $X\in L^{2}(\Omega;\mathbb{R}^{d})$ and $a\in\mathbb{R}^{d}$, $\mathsf{C}(X+c_{a})=X+c_{a}-c_{\mathsf{m}(X)+a}=X+c_{a-\mathsf{m}(X)-a}=X-c_{\mathsf{m}(X)}=\mathsf{C}X$ by the expectation-map paragraph and Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (i). Hence $\phi^{\circ}_{X}(a)=\Phi^{\circ}(X+c_{a})=\Phi(\mathsf{C}X)$ is constant in $a$, a smooth function (claim 2 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions) whose partial derivatives vanish identically (its difference quotients are $0$, Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$), as do their partial derivatives; so $\phi^{\circ}_{X}$ is of class $C^{2}$ with $D^{2}\phi^{\circ}_{X}(0_{\mathbb{R}^{d}})=0_{d}$. Thus $\Phi^{\circ}$ is twice continuously differentiable along translations at every point, and the translation Hessian of $\varphi^{\circ}$ at every $\mu$ is $0_{d}$ (Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (iii)).

(d) and translation invariance. For $\mu$ and $a$, $\varphi^{\circ}((\tau_{a})_{\#}\mu)=\varphi(\overline{(\tau_{a})_{\#}\mu})=\varphi(\bar{\mu})=\varphi^{\circ}(\mu)$ by claim 2. For continuity at $\mu$: given $\varepsilon>0$, property (d) of $\varphi$ gives $\delta>0$ with $|\varphi(\sigma)-\varphi(\bar{\mu})|<\varepsilon$ whenever $W_{2}(\sigma,\bar{\mu})<\delta$ (Continuous Map Between Metric Spaces); if $W_{2}(\nu,\mu)<\delta$ then $W_{2}(\bar{\nu},\bar{\mu})\le W_{2}(\nu,\mu)<\delta$ by claim 2 (and the symmetry of $W_{2}$, The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space (i)), so $|\varphi^{\circ}(\nu)-\varphi^{\circ}(\mu)|<\varepsilon$. Hence $\varphi^{\circ}$ is a test function with the stated gradient and Hessian.