Hamiltonian ODE's on a Space of Deficient Measures

TL;DR

This paper studies Hamiltonian ODEs on measures with mass deficiency due to escape to infinity, introducing a dissipative regularization to analyze the measure's evolution and conditions preventing mass return from infinity.

Contribution

It develops a regularization scheme based on dissipation to handle mass loss in Hamiltonian measure dynamics and establishes conditions for mass escape prevention.

Findings

Dissipative regularization effectively models mass loss.

Conditions identified that prevent mass from returning from infinity.

Analysis advances understanding of measure evolution with mass deficiency.

Abstract

We continue the study (initiated in [1]) of Borel measures whose time evolution is provided by an interacting Hamiltonian structure. Here, the principal focus is the development and advancement of deficency 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.