Weak KAM Theory on the Wasserstein Torus with Multidimensional Underlying Space

The study of asymptotic behavior of minimizing trajectories on the Wasserstein space P(Td) has so far been limited to the case d = 1 as all prior studies heavily relied on the isometric identification of P(T) with a subset of the Hilbert space L2(0,1). There is no known analogue isometric identification when d > 1. In this article we propose a new approach, intrinsic to the Wasserstein space, which allows us to prove a weak KAM theorem on P(Td), the space of probability measures on the torus, for any d ≥ 1. This space is analyzed in detail, facilitating the study of the asymptotic behavior/invariant measures associated with minimizing trajectories of a class of Lagrangians of practical importance. © 2014 Wiley Periodicals, Inc.