  We prove existence and uniqueness results for solutions to a class of optimal
transportation problems with infinitely many marginals, supported on the real
line. We also provide a characterization of the solution with an explicit
formula. We then show that this result implies an infinite dimensional
rearrangement inequality, and use it to derive upper bounds on solutions to
parabolic PDE via the Feynman-Kac formula. This result can be used to provide
model independent bounds on the prices of certain derivatives in mathematical
finance. As an another application, we obtain refined phase-space bounds in
quantum physics.
