Stochastic Wasserstein Hamiltonian Flows

In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with L-2-Wasserstein metric tensor, via the Wong-Zakai approximation. We begin our investigation by showing that the stochastic Euler-Lagrange equation, regardless it is deduced from either the variational principle or particle dynamics, can be interpreted as the stochastic kinetic Hamiltonian flows in Wasserstein manifold. We further propose a novel variational formulation to derive more general stochastic Wasserstein Hamiltonian flows, and demonstrate that this new formulation is applicable to various systems including the stochastic Schrodinger equation, Schrodinger equation with random dispersion, and Schrodinger bridge problem with common noise.

Automated summary

In this paper, the stochastic Hamiltonian flow in Wasserstein manifold is studied via the Wong-Zakai approximation. It is shown that the stochastic Euler-Lagrange equation can be interpreted as the stochastic kinetic Hamiltonian flows in Wasserstein manifold, regardless of its derivation from either the variational principle or particle dynamics. A novel variational formulation is proposed to derive more general stochastic Wasserstein Hamiltonian flows, and its applicability to various systems including the stochastic Schrodinger equation, Schrodinger equation with random dispersion, and Schrodinger bridge problem with common noise is demonstrated.