Optimal transportation with infinitely many marginals

We formulate and study an optimal transportation problem with infinitely many marginals; this is a natural extension of the multi-marginal problem studied by Gangbo and Świȩch (1998) [15], . We prove results on the existence, uniqueness and characterization of the optimizer, which are natural extensions of the results in Gangbo and Świȩch (1998) [15], . The proof relies on a relationship between this problem and the problem of finding barycenters in the Wasserstein space, a connection first observed for finitely many marginals by Agueh and Carlier (2011) [1].