The First Marginal of a Probability Measure on a Product: Finite Second Moment, the Wasserstein Contraction, and the Gradient Plan

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $k\in\mathbb{N}$ satisfy $1\le k$ and let $\gamma,\gamma'\in\mathcal{P}_{2}(\mathbb{R}^{d+k})$, with first marginals $\gamma^{(1)},\gamma'^{(1)}\in\mathcal{P}(\mathbb{R}^{d})$ formed with the coordinate projection $\mathrm{pr}^{d,k}_{1}:\mathbb{R}^{d+k}\to\mathbb{R}^{d}$. Plans, the variables $x,p$ and the integral notation are as fixed there, and $L^{2}(\gamma;\mathbb{R}^{d+k})$ is as in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii). Then the following hold.

1. (The first marginal has finite second moment)(i) $\gamma^{(1)}\in\mathcal{P}_{2}(\mathbb{R}^{d})$, with $M_{2}(\gamma^{(1)})\le M_{2}(\gamma)$.

2. (The Wasserstein contraction)(ii)

$$W_{2}\bigl(\gamma^{(1)},\gamma'^{(1)}\bigr)\le W_{2}(\gamma,\gamma').$$

3. (The gradient plan of a vector field)(iii) Let $\eta\in L^{2}(\gamma;\mathbb{R}^{d+k})$ and let

$$\Xi^{\eta}=\bigl(\mathrm{pr}^{d,k}_{1},\ \mathrm{pr}^{d,k}_{1}\circ\eta\bigr):\mathbb{R}^{d+k}\to\mathbb{R}^{d+d}$$

be the pairing formed from a representative of $\eta$, a Borel map. Then $\Xi^{\eta}_{\#}\gamma$ is a plan with first marginal $\gamma^{(1)}$, it does not depend on the representative of $\eta$ chosen, and

$$\int\lVert p\rVert^{2}\,d\bigl(\Xi^{\eta}_{\#}\gamma\bigr)\le\lVert\eta\rVert_{\gamma}^{2}.$$