  We study gradient drift-diffusion processes on a probability simplex set with
finite state Wasserstein metrics, namely the Wasserstein common noise. A fact
is that the Kolmogorov transition equation of finite reversible Markov jump
processes forms the gradient flow of entropy in finite state Wasserstein space.
This paper proposes to perturb finite state Markov jump processes with
Wasserstein common noises and formulate stochastic reversible Markov jumping
processes. We also define a Wasserstein Q-matrix for this stochastic Markov
jumping process. We then derive the functional Fokker-Planck equation in
probability simplex, whose stationary distribution is a Gibbs distribution of
entropy functional in a simplex set. Finally, we present several examples of
Wasserstein drift-diffusion processes on a two-point state space.
