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We prove existence and uniqueness results for solutions to a class of optimaltransportation problems with infinitely many marginals, supported on the realline. We also provide a characterization of the solution with an explicitformula. We then show that this result implies an infinite dimensionalrearrangement inequality, and use it to derive upper bounds on solutions toparabolic PDE via the Feynman-Kac formula. This result can be used to providemodel independent bounds on the prices of certain derivatives in mathematicalfinance. As an another application, we obtain refined phase-space bounds inquantum physics.

On a class of optimal transportation problems with infinitely many marginals