Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to
$$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable mappings }T: X \rightarrow Y \text{ and } T\_\#\mu = \nu$$
has a relatively straightforward interpretation: We try to find a measurable map $T$ that minimizes the cost to move mass from $x$ to $T(x)$. Now, the Kantorovich problem to
$$ \text{minimize} \quad \int c(x,y)\pi(dx,dy) \quad \text{over couplings } \pi \text{ with first and second marginals } \mu \text{ and } \nu \text{, respectively,}$$
I find to be much harder to interpret as a real `mass transfer' problem; If $\pi^\star$ is an optimal coupling to the Kantorovich problem, what does $\pi^\star$ tell me where how much mass really goes? How do I interpret the Kantorovich problem?



How to interpret couplings in optimal transport?
Of the mass $\mu(A)$ in $A$ a fraction $\pi(A \times B)$ is transported to $B$, so you can think of this as a randomized transport map. A basic example to think of is $\mu=\delta\_0$ and $\nu=(\delta\_1+\delta\_{-1})/2$. Half the mass at 0 is sent to 1 and half is sent to -1. You can get a better intuition from reading more about construction of couplings:


[1] Lindvall, Torgny. Lectures on the coupling method. Courier Corporation, 2002.


[2] Thorisson, Hermann. "Coupling methods in probability theory." Scandinavian journal of statistics (1995): 159-182.


