Weak KAM Theory on the Wasserstein Torus with Multidimensional Underlying Space

The study of asymptotic behavior of minimizing trajectories on the Wasserstein space ��(��^d) has so far been limited to the case d = 1 as all prior studies heavily relied on the isometric identification of ��(��) 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 ��(��^d), 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.