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Let $\mu$, $\nu$ and $\gamma$ be three probability measures on $\mathbb R$. Consider the optimisation problem as follows: $$\inf_{(X,Y,Z)}~ \mathbb E\big[|Y-Z|^2\big],$$ where the inf is taken overa …

Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below: $$W_2(\mu,\nu)^2 \quad := \quad \inf_{ …

I've a solution but it's not perfectly satisfying. Assume that $$V~~~:=~~~\int |x|^pd\mu(x)~+~\int |x|^pd\nu(x)~~~<~~~+\infty$$ for some fixed $p>1$. It follows from Thought 1 that, there exists $f …

Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$. My …

Let $K=[0,1]^2$ be a square and $p\in (0,1)$ be a fixed number. We define a map $F: K^2\to K^2$ as follows. For $(x_1,y_1), (x_2,y_2)\in K$, it follows by a straightforward computation that there e …

Thank @AnthonyQuas for the reply. I claim that this is not an answer, but just a question related to the last remark by @AnthonyQuas Consider one case where $\Delta x:=x_2-x_1>0$, $\Delta y:=y_2-y_1> …