Hamiltonian ODE’s in the Wasserstein space of probability measures

In this paper we consider a Hamiltonian H on P2(R), 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 +∇ · (Jdvt μt) = 0, where Jd is the canonical symplectic matrix, μ0 is prescribed, vt is a tangent vector to P2(R) at μt , and belongs 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 furthermore assume that H is λ–convex, proper and lower semicontinuous on P2(R), we prove that the Hamiltonian is preserved along any solution of our evolutive system: H(μt) = H(μ0). c © 2000 Wiley Periodicals, Inc.