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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, with the Wasserstein distance $W_{2}$, couplings $\Pi(\cdot,\cdot)$ and their quadratic cost $I$ as fixed there, and let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ be a test function, with gradient map $\nabla\psi:\mathbb{R}^{d}\to\mathbb{R}^{d}$, whose class in $L^{2}(\mu;\mathbb{R}^{d})$ has norm $\lVert\nabla\psi\rVert_{\mu}$. Let $\mathrm{id}$ be the identity map of $\mathbb{R}^{d}$, and for $s\in\mathbb{R}$ let $G_{s}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the map

$$G_{s}(x)=x+s\,\nabla\psi(x)\qquad(x\in\mathbb{R}^{d}),$$

abbreviated $G_{s}=\mathrm{id}+s\,\nabla\psi$, the sum and the scalar multiple of points of $\mathbb{R}^{d}$ being those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i). Fix $t\in\mathbb{R}$ and write $t^{2}=t\cdot t$. Push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), and $(\mathrm{id},G_{t}):\mathbb{R}^{d}\to\mathbb{R}^{d+d}$ is the pairing, both notations being available for $G_{t}$ once claim 1 has shown it to be Borel. Then the following hold.

1. (Borel, finite second moment, and the case $t=0$)(i) The map $G_{t}$ is Borel and its push-forward $(G_{t})_{\#}\mu$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$; and $G_{0}=\mathrm{id}$, so that $(G_{0})_{\#}\mu=\mu$.

2. (The diagonal coupling)(ii) $(\mathrm{id},G_{t})_{\#}\mu\in\Pi\bigl(\mu,(G_{t})_{\#}\mu\bigr)$, and

$$I\bigl((\mathrm{id},G_{t})_{\#}\mu\bigr)=t^{2}\,\lVert\nabla\psi\rVert_{\mu}^{2}.$$

3. (Wasserstein bound)(iii)

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