  In this work we study a modification of the Monge-Kantorovich problem taking
into account path dependence and interaction effects between particles. We
prove existence of solutions under mild conditions on the data, and after
imposing stronger conditions, we characterize the minimizers by relating them
to an auxiliary Monge-Kantorovich problem of the more standard kind. With this
notion of how particles interact and travel along paths, we produce a dual
problem. The main novelty here is to incorporate an interaction effect to the
optimal path transport problem. This covers for instance, $N$-body dynamics
when the underlying measures are discrete. Lastly, our results include an
extension of Brenier's theorem on optimal transport maps.
