Proof of 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

Each result cited is universally quantified over the data in its own statement. Points of $\mathbb{R}^{d}$ are read as $d$-tuples by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i), the $i$th component of $x$ being $x_{i}$; the sum and scalar multiple of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i) are the sum of points and the scalar multiple, and the origin $0_{\mathbb{R}^{d}}$ is the zero vector of the real vector space $\mathbb{R}^{d}$ by claim 1 of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space.

Step 1: $G_{t}$ is Borel. By Sum of Points of $\mathbb{R}^n$ and Scalar Multiple of a Point of $\mathbb{R}^n$, the $i$th component of $G_{t}(x)=x+t\,\nabla\psi(x)$ is $x_{i}+t\,(\nabla\psi(x))_{i}$ for every $i\in[d]$. The map $x\mapsto x_{i}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{d})$ and $\mathcal{B}(\mathbb{R})$ by claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, whose claims are in force for $\mathcal{B}(\mathbb{R}^{d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i). The map $\nabla\psi$ is Borel by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure (i), so each component $x\mapsto(\nabla\psi(x))_{i}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{d})$ and $\mathcal{B}(\mathbb{R})$ by the componentwise criterion, claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets. Hence each component $x\mapsto x_{i}+t\,(\nabla\psi(x))_{i}$ of $G_{t}$ is measurable by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, and $G_{t}$ is Borel by claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets again. Consequently $(G_{t})_{\#}\mu\in\mathcal{P}(\mathbb{R}^{d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), and the pairing $(\mathrm{id},G_{t})$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), the identity map being Borel by the preamble of 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.

Step 2: claim 2. 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 (iv), applied with $S=\mathrm{id}$ and $T=G_{t}$, the push-forward $\pi=(\mathrm{id},G_{t})_{\#}\mu$ belongs to $\Pi\bigl(\mathrm{id}_{\#}\mu,(G_{t})_{\#}\mu\bigr)$ and

$$I(\pi)=\int_{\mathbb{R}^{d}}\lVert x-G_{t}(x)\rVert^{2}\,\mu(dx).$$

Here $\mathrm{id}_{\#}\mu=\mu$, since $\mathrm{id}_{\#}\mu(B)=\mu(\mathrm{id}^{-1}(B))=\mu(B)$ for every $B\in\mathcal{B}(\mathbb{R}^{d})$ by the formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv); so $\pi\in\Pi\bigl(\mu,(G_{t})_{\#}\mu\bigr)$. For every $x\in\mathbb{R}^{d}$, by 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 twice and then its claim 5,

$$\lVert x-G_{t}(x)\rVert=d_{E}\bigl(x,x+t\,\nabla\psi(x)\bigr)=\lVert t\,\nabla\psi(x)\rVert=|t|\,\lVert\nabla\psi(x)\rVert ,$$

and squaring, using commutativity and associativity of multiplication in the field $\mathbb{R}$ and $|t|^{2}=t^{2}$ (claim 1 of Nonnegativity of Squares in an Ordered Field),

$$\lVert x-G_{t}(x)\rVert^{2}=t^{2}\,\lVert\nabla\psi(x)\rVert^{2}.$$

The function $x\mapsto\lVert\nabla\psi(x)\rVert^{2}$ is Borel with $\int_{\mathbb{R}^{d}}\lVert\nabla\psi\rVert^{2}\,d\mu<\infty$ by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure (i), and it is nonnegative, as is $t^{2}$, by claim 2 of Nonnegativity of Squares in an Ordered Field; so by the homogeneity of the integral of nonnegative functions, claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion,

$$I(\pi)=\int_{\mathbb{R}^{d}}t^{2}\,\lVert\nabla\psi\rVert^{2}\,d\mu=t^{2}\int_{\mathbb{R}^{d}}\lVert\nabla\psi\rVert^{2}\,d\mu=t^{2}\,\lVert\nabla\psi\rVert_{\mu}^{2},$$

the last equality being the formula for the norm of $L^{2}(\mu;\mathbb{R}^{d})$ in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii), the square of a nonnegative square root being the radicand by Existence and Uniqueness of the Nonnegative Square Root. This proves claim 2; in particular $I(\pi)<\infty$.

Step 3: claim 1. Since $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi\bigl(\mu,(G_{t})_{\#}\mu\bigr)$ has $I(\pi)<\infty$ by Step 2, the converse part of 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) gives $(G_{t})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. For the map $G_{0}$ of the statement and every $x\in\mathbb{R}^{d}$, $0\,\nabla\psi(x)=0_{\mathbb{R}^{d}}$ by claim 3 of Elementary Identities in a Vector Space, so $G_{0}(x)=x+0_{\mathbb{R}^{d}}=x$ by the zero-vector property of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space; thus $G_{0}=\mathrm{id}$ and $(G_{0})_{\#}\mu=\mathrm{id}_{\#}\mu=\mu$ as in Step 2. Together with Step 1 this proves claim 1.

Step 4: claim 3. By The Quadratic Wasserstein Distance on Euclidean Space (i), $W_{2}\bigl(\mu,(G_{t})_{\#}\mu\bigr)$ is a nonnegative real number with

$$W_{2}\bigl(\mu,(G_{t})_{\#}\mu\bigr)^{2}\le I(\pi)=t^{2}\,\lVert\nabla\psi\rVert_{\mu}^{2}=\bigl(|t|\,\lVert\nabla\psi\rVert_{\mu}\bigr)^{2},$$

by claim 2 and the same rearrangement as in Step 2. The number $|t|\,\lVert\nabla\psi\rVert_{\mu}$ is nonnegative: $0\le|t|$ by claim 1 of Properties of the Absolute Value in an Ordered Field, $0\le\lVert\nabla\psi\rVert_{\mu}$ as the nonnegative square root of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii) and Existence and Uniqueness of the Nonnegative Square Root, so $|t|\cdot0\le|t|\,\lVert\nabla\psi\rVert_{\mu}$ by claim 5 of Elementary Arithmetic in an Ordered Field, and $|t|\cdot0=0$ by claim 1 of Zero Products and Elementary Identities in a Field. Hence claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, applied to the nonnegative numbers $W_{2}\bigl(\mu,(G_{t})_{\#}\mu\bigr)$ and $|t|\,\lVert\nabla\psi\rVert_{\mu}$, gives $W_{2}\bigl(\mu,(G_{t})_{\#}\mu\bigr)\le|t|\,\lVert\nabla\psi\rVert_{\mu}$. $\blacksquare$