Hamiltonian ODEs in the Wasserstein space of probability measures

In this paper we consider a Hamiltonian H on 2 (ℝ 2 d ), the set of probability measures with finite quadratic moments on the phase space ℝ 2 d = ℝ d × ℝ d , which is a metric space when endowed with the Wasserstein distance W 2 . We study the initial value problem d μ t / dt + ∇ · ( d v t μ t ) = 0, where d is the canonical symplectic matrix, μ 0 is prescribed, and v t is a tangent vector to 2 (ℝ 2 d ) at μ t , belonging to ∂ H (μ t ), the subdifferential of H at μ t . Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ 0 is absolutely continuous. It ensures that μ t remains absolutely continuous and v t = ∇ H (μ t ) is the element of minimal norm in ∂ H (μ t ). The second method handles any initial measure μ 0 . If we further assume that H is λ‐convex, proper, and lower‐semicontinuous on 2 (ℝ 2 d ), we prove that the Hamiltonian is preserved along any solution of our evolutive system, H (μ t ) = H (μ 0 ). © 2007 Wiley Periodicals, Inc.