Hamiltonian ODEs in the Wasserstein space of probability measures

✍ Scribed by Luigi Ambrosio; Wilfrid Gangbo

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 299 KB
• Volume: 61
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.20188

✦ Synopsis

In this paper we consider a Hamiltonian H on P 2 (R 2d ), the set of probability measures with finite quadratic moments on the phase space R 2d = R d × R d , which is a metric space when endowed with the Wasserstein distance W 2 . We study the initial value problem dµ t /dt +∇•(J d v t µ t ) = 0, where J d is the canonical symplectic matrix, µ 0 is prescribed, and v t is a tangent vector to P 2 (R 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 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 P 2 (R 2d ), we prove that the Hamiltonian is preserved along any solution of our evolutive system, H (µ t ) = H (µ 0 ).