  We present a definition of stochastic Hamiltonian process on finite graph via
its corresponding density dynamics in Wasserstein manifold. We demonstrate the
existence of stochastic Hamiltonian process in many classical discrete
problems, such as the optimal transport problem, Schr\"odinger equation and
Schr\"odinger bridge problem (SBP). The stationary and periodic properties of
Hamiltonian processes are also investigated in the framework of SBP.
