Moreau-Yosida approximation and convergence of Hamiltonian systems on Wasserstein space

TL;DR

This paper investigates the stability and convergence of Hamiltonian systems on Wasserstein space by using Moreau-Yosida regularization of the Hamiltonian and demonstrating solution convergence as regularization diminishes.

Contribution

It introduces a framework for regularizing Hamiltonians on Wasserstein space and proves convergence of solutions, extending stability analysis in this geometric setting.

Findings

Solutions of regularized systems converge to the original system as regularization parameter tends to zero.

Provides sufficient conditions on Hamiltonian functions for convergence.

Establishes a stability property for Hamiltonian systems on Wasserstein space.

Abstract

In this paper, we study the stability property of Hamiltonian systems on the Wasserstein space. Let $H$ be a given Hamiltonian satisfying certain properties. We regularize $H$ using the Moreau-Yosida approximation and denote it by $H_\tau.$ We show that solutions of the Hamiltonian system for $H_\tau$ converge to a solution of the Hamiltonian system for $H$ as $\tau$ converges to zero. We provide sufficient conditions on $H$ to carry out this process.