Hamiltonian ODEs in the Wasserstein space of probability measures

In this paper we consider a Hamiltonian H on ??₂(?^2d), the set of probability measures with finite quadratic moments on the phase space ?^2d = ?^d × ?^d, which is a metric space when endowed with the Wasserstein distance W₂. We study the initial value problem dμ_t/dt + ? · (??_d v _tμ_t) = 0, where ??_d is the canonical symplectic matrix, μ₀ is prescribed, and v _t is a tangent vector to ??₂(?^2d) 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 μ₀ 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 μ₀. If we further assume that H is λ‐convex, proper, and lower‐semicontinuous on ??₂(?^2d), we prove that the Hamiltonian is preserved along any solution of our evolutive system, H(μ_t) = H(μ₀). © 2007 Wiley Periodicals, Inc.