The Joint Law of Two Classes of Square-Integrable Random Vectors: Marginals, Cost, and Invariance Under Shifts by a Vector Field

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $X,V,X',V'\in L^{2}(\Omega;\mathbb{R}^{d})$, the space of classes of square-integrable random vectors in $\mathbb{R}^{d}$ with its norm $\lVert\cdot\rVert_{L^{2}}$ and the law $\mathcal{L}(X)$ of a class, and let $M_{2}$ be the second moment of that clause. Push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), and $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ are the coordinate projections of $\mathbb{R}^{d+d}$. For representatives of $X$ and of $V$, their pairing $(X,V)$ is a random vector in $\mathbb{R}^{d+d}$ by Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (v), with the law of that definition. Couplings and their quadratic cost $I$ are as fixed in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i), and for $\eta$ in the space $L^{2}(\mathcal{L}(X);\mathbb{R}^{d})$ the composition $\eta\circ X\in L^{2}(\Omega;\mathbb{R}^{d})$ is that of that clause. Then the following hold.

1. (The joint law)(i) The law of the pairing $(X,V)$ of representatives of $X$ and of $V$ does not depend on the representatives chosen; it is called the joint law of $X$ and $V$ and written $\mathcal{L}(X,V)$. It belongs to $\mathcal{P}_{2}(\mathbb{R}^{d+d})$, and

$$M_{2}\bigl(\mathcal{L}(X,V)\bigr)=\lVert X\rVert_{L^{2}}^{2}+\lVert V\rVert_{L^{2}}^{2}.$$

2. (Marginals and cost)(ii) $(\mathrm{pr}_{1})_{\#}\mathcal{L}(X,V)=\mathcal{L}(X)$ and $(\mathrm{pr}_{2})_{\#}\mathcal{L}(X,V)=\mathcal{L}(V)$; thus $\mathcal{L}(X,V)$ is a coupling of $\mathcal{L}(X)$ and $\mathcal{L}(V)$, and its quadratic cost is

$$I\bigl(\mathcal{L}(X,V)\bigr)=\lVert X-V\rVert_{L^{2}}^{2}.$$

3. (Invariance under shifts by a vector field)(iii) Suppose that $\mathcal{L}(X,V)=\mathcal{L}(X',V')$. Then $\mathcal{L}(X)=\mathcal{L}(X')$, so that $L^{2}(\mathcal{L}(X);\mathbb{R}^{d})$ and $L^{2}(\mathcal{L}(X');\mathbb{R}^{d})$ are the same space and $\eta\circ X'$ is defined for $\eta$ in it; and for every $\eta\in L^{2}(\mathcal{L}(X);\mathbb{R}^{d})$ and every $t\in\mathbb{R}$,

$$\mathcal{L}\bigl(X,\ V+t\,\eta\circ X\bigr)=\mathcal{L}\bigl(X',\ V'+t\,\eta\circ X'\bigr).$$