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I am fairly sure this is not the case (unless I am missing something in the definition of $T$) Consider the special case where $T$ is invertible with inverse $T_{\rm inv}$. Then, it holds for any test …

As already mentioned in the comments, this is really a standard result, see, e.g., Lemma 3.1. in Kallenberg's book.

First, I interpret your condition that "$X$ has no isolated points" in the following ways: First, every ball $B(x, \varepsilon)$ has non-empty interior. This means, in particular, that we can find arb …

There is a bound, the coupling $(\xi, a \xi)$ leads to the upper bound \begin{align*} W_p(\xi, a \xi)^p &\leq \mathbb{E}[|\xi - a\xi|^p] = |(1-a)|^p \mathbb{E}[|\xi|^p]\\ \Rightarrow ~~~~ W_p(\xi, a\x …

The following question came up when trying to numerically solve some variants of the Gromov-Wasserstein distance. Setting: Let $(X_1, d_1), (X_2, d_2)$ be two compact, separable and complete metric s …

There are various papers where this question occurs. I guess a paper which directly covers the case you are interested in is https://arxiv.org/pdf/1901.07407.pdf . Note that here, the marginals don't …

I'll preface this by saying that this is not a complete answer. First, there is a very nice Python package called POT which has calculation of Wasserstein-Gromov distance included (for discrete measu …

Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ endo …