# Questions tagged [wasserstein-distance]

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### Some stability and estimate of the optimal transport map (Brenier map)

Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
• 72
1 vote
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### Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?

Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
• 617
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### Some continuity issues of the optimal transport map (Brenier map)

Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
• 72
1 vote
68 views

### Gradient flows: evolution of geodesics

I’m trying to understand if, when I move the marginals of a Wasserstein geodesic along a contractive flow, the geodesic between the new probability measures is “near” to the geodesic connecting the ...
• 11
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### Perturbation of Wasserstein distance: looking for references

I am doing readings related to Optimal transport which is new to me and I often encounter the following statement regarding a sort of derivative of the Wasserstein distance: $u$ and $v$ be two ...
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• 5,069
1 vote
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### Bounding $2$-Wasserstein distance and the $L^1$ distance

My questions come from the paper Logarithmic Sobolev inequalities for some nonlinear PDE’s written by F. Malrieu (May 2001) where author omitted a good amount of details to be filled. Suppose that $W$ ...
• 680
252 views

### 2-Wasserstein metric on convolution of probability distributions

I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$ with finite second moments, and $W_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question ...
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### Gradient of Wasserstein distance in the sense of Otto's calculus

I am learning the idea of "gradient" of a functional in Otto's calculus. It is defined as follows. Suppose the space we are thinking about is $(\mathcal{P}_{2,AC}(\mathbb{R}^d),W_2)$, the ...
• 593
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### Convergence of empirical measures in Wasserstein distance

Let $X_1, X_2, \ldots$ be iid random variables with common distribution $\gamma$, the standard Gaussian distribution on $\mathbb {R}$, and let $\mu_n = \frac 1n \sum_{i=1}^n \delta_{X_i}$, $n \geq 1$, ...
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### Wasserstein distance and Monge-Kantorovich-Rubinstein duality

The definition of Wasserstein $p$-distance between two measures $\mu$ and $\nu$ on a Polish space $X$ is given by  W_p(\mu, \nu)^p = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...
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### Upper bound total variation by Wasserstein distance for continuous distance

I am reading the survey of the relationships between metrics of distributions (see https://arxiv.org/pdf/math/0209021.pdf for the paper). The general results show that for general distributions, we ...
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Edit: Changed from "Hausdorff" to "metric" spaces. Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, ...
This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies. Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...