Optimal Transportation Problems for Vector Measures

Given two probability measure spaces $\left(X,\mu\right),\left(Y,\nu\right)$ and a cost function $c\left(x,y\right)$ on their product space, the optimal transport problem is to find a map $T:X\rightarrow Y$ that minimizes the total cost $\int_{X}c\left(x,Tx\right)\mu\left(dx\right)$ among all maps that push $\mu$ to $\nu$ (i.e. $T_{\#}\mu=\nu$). In this talk, we suggest a generalization for this problem using vector measures instead of scalar measures, which arises by adding additional contraints of the form $T_{\#}\lambda=\gamma$. We discuss the physical motivation behind such problems and present a new duality theorem.