In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with L2-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 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 Wassersetin Hamiltonian flows, and demonstrate that this new formulation is applicable to various systems including the stochastic Schrödinger equation, Schrödinger equation with random dispersion, and Schrödinger bridge problem with common noise.
Mathematics Subject Classification (2000) Primary 58B20, Secondary 35R60, 35Q41, 35Q83, 65M75