Hamiltonian ODEs in the Wasserstein space of probability measures

In this paper we consider a Hamiltonian H on P2(R2d), the set of probability measures with finite quadratic moments on the phase space R2d = Rd × Rd, which is a metric space when endowed with the Wasserstein distance W2. We study the initial value problem dμt/dt + ∇ · (Jdvtμt) = 0, where Jd is the canonical symplectic matrix, μ0 is prescribed, and vt is a tangent vector to P2(R2d) at μt, belonging to ∂H(μt), the subdifferential of H at μt. Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ0 is absolutely continuous. It ensures that μt remains absolutely continuous and vt = ∇H(μt) is the element of minimal norm in ∂H(μt). The second method handles any initial measure μ0. If we further assume that H is λ‐convex, proper, and lower‐semicontinuous on P2(R2d), we prove that the Hamiltonian is preserved along any solution of our evolutive system, H(μt) = H(μ0). © 2007 Wiley Periodicals, Inc.