2022-11-21

Let X be a separable Banach space. The p-Wasserstein space on X is defined as
${W}_{p}\left(X\right)=\left\{\mu \in P\left(X\right):{\int }_{X}||x|{|}^{p}d\mu \left(x\right)<\mathrm{\infty }\right\},p\ge 1$

Antwan Wiley

Expert

Step 1
For $\mathcal{X}={\mathbb{R}}^{d}$, you can choose the Euclidean norm or the ${\ell }^{p}$ norm, which is equivalent, that is, there exists constants ${C}_{p,d}$ and ${C}_{p,d}^{\prime }$ such that ${C}_{p,d}{\left(\sum _{k=1}^{d}|{x}_{k}{|}^{p}\right)}^{1/p}⩽\sqrt{\sum _{k=1}^{d}{x}_{k}^{2}}⩽{C}_{p,d}^{\prime }{\left(\sum _{k=1}^{d}|{x}_{k}{|}^{p}\right)}^{1/p}$. With this in mind, we derive that

Step 2
where ${\mu }_{i}$ is the i-marginal: ${\mu }_{i}\left(B\right)=\mu \left({\mathbb{R}}^{i-1}×B×{\mathbb{R}}^{d-i}\right)$

Do you have a similar question?