Wasserstein Hamiltonian flow with common noise on graph

TL;DR

This paper develops a stochastic Wasserstein Hamiltonian flow framework with common noise on graphs, proving local and global existence of solutions, and establishing connections to nonlinear Schrödinger equations and optimal control problems.

Contribution

It introduces a novel formulation of stochastic Wasserstein Hamiltonian flow on graphs and proves existence, uniqueness, and optimal control solutions within this framework.

Findings

Established local and global existence of solutions.

Connected stochastic flow to nonlinear Schrödinger equations.

Proved existence of minimizers for control problems with noise.

Abstract

We study the Wasserstein Hamiltonian flow with a common noise on the density manifold of a finite graph. Under the framework of stochastic variational principle, we first develop the formulation of stochastic Wasserstein Hamiltonian flow and show the local existence of a unique solution. We also establish a sufficient condition for the global existence of the solution. Consequently, we obtain the global well-posedness for the nonlinear Schr\"odinger equations with common noise on graph. In addition, using Wong-Zakai approximation of common noise, we prove the existence of the minimizer for an optimal control problem with common noise. We show that its minimizer satisfies the stochastic Wasserstein Hamiltonian flow on graph as well.