WASSERSTEIN HAMILTONIAN FLOW WITH COMMON NOISE ON GRAPH

We study the Wasserstein Hamiltonian flow with a common noise on the density manifold of a finite graph. Under the framework of the 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 Schro\dinger equations with common noise on a 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 a graph as well.

Automated summary

This article studies the Wasserstein Hamiltonian flow with common noise on the density manifold of a finite graph. Under the framework of the stochastic variational principle, the formulation of stochastic Wasserstein Hamiltonian flow is developed, and the local existence of a unique solution is shown. A sufficient condition for the global existence of the solution is also established. Consequently, the global well-posedness for the nonlinear Schroedinger equations with common noise on a graph is obtained. Additionally, the existence of the minimizer for an optimal control problem with common noise is proved using Wong-Zakai approximation, and it is shown that the minimizer satisfies the stochastic Wasserstein Hamiltonian flow on a graph as well.