Weak KAM Theory on the Wasserstein Torus with Multidimensional Underlying Space

The study of asymptotic behavior of minimizing trajectories on the Wasserstein space P(T-d) has so far been limited to the case d=1 as all prior studies heavily relied on the isometric identification P(T) of with a subset of the Hilbert space L-2(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(T-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. (c) 2014 Wiley Periodicals, Inc.