Quadratic optimal transportation problem with a positive semi definite structure on the cost function

Optimal transportation problem seeks for a coupling $\pi$ of two probability measures $\mu$ and $\nu$ which minimize the total cost $\int cd\pi$, which is linear in $\pi$. In this paper, we introduce a variation of optimal transportation problem which we call quadratic transportation problem that considers a total cost $\iint cd\pi d\pi$ which is quadratic in $\pi$. We compare this problem with other variations of optimal transportation problem, and prove some properties of the solutions to the problem. Then, we introduce squared cost function, which let us consider the total cost $\iint cd\pi d\pi$ as a positive semi-definite bilinear operator on probability measures, and show Kantorovich duality formula when we have a squared cost function.