Communications in Mathematical Sciences

Volume 11 (2013)

Number 1

Hamiltonian ODEs on a space of deficient measures

Pages: 1 – 31



L. Chayes (Department of Mathematics, University of California at Los Angeles)

W. Gangbo (Department of Mathematics, Georgia Institute of Technology, Atlanta, Ga., U.S.A.)

H. K. Lei (Department of Mathematics, California Institute of Technology, Pasadena, Cal., U.S.A.)


We continue the study (initiated in [L. Ambrosio and W. Gangbo, Commun. Pure Appl. Math., 61, 18–53, 2007] of Borel measures whose time evolution is provided by an interacting Hamiltonian structure. Here, the principal focus is the development and advancement of deficiency in the measure caused by displacement of mass to infinity in finite time. We introduce—and study in its own right—a regularization scheme based on a dissipative mechanism which naturally degrades mass according to distance traveled (in phase space). Our principal results are obtained based on some dynamical considerations in the form of a condition which forbids mass to return from infinity.


infinite dimensional Hamiltonian, ODEs on measure spaces, Wasserstein metric

2010 Mathematics Subject Classification

35Qxx, 37Kxx, 49-xx, 53Dxx

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