Definition (Wasserstein metric)

Let $(\mathcal{X},d)$ be a Polish metric space, and let $1\le p<\infty$. For any two Probability Measures $\mu,\nu$ on $\mathcal{X}$ the Wasserstein Metric of order $p$ between $\mu$ and $\nu$ is defined by the formula $$\begin{align*} W_{p}(\mu,\nu)&=\left( \inf_{\pi \in\Pi(\mu,\nu)}\int\limits _{\mathcal{X}\times \mathcal{X}}d(x,y)^{p} \, d\pi(x,y) \right)^{1/p}\\ &=\inf\left \{ [E_{}\left[ d(X,Y)^{p} \right]^{1/p},\,\text{law}(X)=\mu,\,\text{law}(Y)=\nu \right\} \end{align*}$$

Definition (Wasserstein-1)

The Wasserstein metric of order 1 for any two Measures $\mu,\nu \in\mathcal{P}(\mathbb{X})$ is defined as $$W_{1}(\mu,\nu)=\inf_{\eta \in\mathcal{H}(\mu,\nu)}\int\limits _{\mathbb{X}\times \mathbb{X}}\eta(dx,dy)|x-y|$$where $\mathcal{H}(\mu,\nu)$ denotes the set of probability measures on $\mathbb{X}\times \mathbb{X}$ with the first marginal $\mu$ and second marginal $\nu$. Furthermore we equivalently have $$W_{1}(\mu,\nu)=\sup_{\lVert f \rVert _{Lip}\le 1}\left|\int\limits f \, d\mu -\int\limits f \, d\nu \right|$$where $$\lVert f \rVert _{Lip}:=\sup_{x\not=y} \frac{f(x)-f(y)}{d_{\mathbb{X}}(x,y)}$$and $d_{\mathbb{X}}$ is the metric on $\mathbb{X}$.

Definition (Wasserstein convergence)

Let $(\mu_{n})_{n\in\mathbb{N}}\subset \mathcal{P}(\mathbb{X}),\mu \in\mathcal{P}(\mathbb{X})$. We say that $\mu_{n}\to \mu$ in $W_{1}(\mu_{n},\mu)$ (the Wasserstein metric) if $$W_{1}(\mu_{n},\mu)\to0\text{ as }n\to \infty$$

Definition (Wasserstein space)

With the same convention defined in the Wasserstein metric the Wasserstein space of order $p$ is defined as $$P_{p}(\mathcal{X}):=\left\{ \mu \in\mathcal{P}(\mathcal{X})\middle| \int\limits _{\mathcal{X}}d(x_{0},x)^{p} \, \mu(dx)<+\infty \right \}\quad x_{0}\in\mathcal{X}$$where $x_{0}\in\mathcal{X}$ is arbitrary. This space does not depend on the choice of the point $x_{0}$. Then $W_{p}$ defines a finite Metric on $P_{p}(\mathcal{X})$.

Definition (Weak convergence in Wasserstein space)

Let $(\mathcal{X},d)$ be a Polish space, and $1\le p<\infty$. Let $(\mu_{k})_{k\in\mathbb{N}}\subset P_{p}(\mathcal{X})$ and $\mu \in P_{p}(\mathcal{X})$ where $P_{p}(\mathcal{X})$ is the . Then $\mu_{k}$ is said to converge weakly in $P_{p}(\mathcal{X})$ if any one of the following properties is satisfied for any $x_{0}\in\mathcal{X}$:

1. $\mu_{k}\to \mu$ and $$\int\limits d(x_{0},x)^{p} \, d\mu_{k}(x)\to \int\limits d(x_{0},x)^{p} \, d\mu(x)$$
2. $\mu_{k}\to \mu$ and $$\limsup_{ k \to \infty } \int\limits d(x_{0},x)^{p} \, d\mu_{k}(x)\le \int\limits d(x_{0},x)^{p} \, d\mu(x)$$
3. $\mu_{k}\to \mu$ and $$\lim_{ R \to \infty } \limsup_{ k \to \infty } \int\limits_{d(x_{0},x)\ge R} d(x_{0},x)^{p} \, d\mu_{k}(x)=0$$
4. For all continuous functions $\varphi$ with $|\varphi(x)|\le C(1+d(x_{0},x)^{p})$, $C\in\mathbb{R}$, one has $$\int\limits \varphi(x) \, d\mu_{k}(x)\to \int\limits \varphi(x) \, d\mu(x)$$