Stochastic Wasserstein Hamiltonian Flows

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