Stationary quantum Markov process for the Wigner function

Many stochastic models have been investigated for quantum mechanics because of its stochastic nature. For example, Cohendet et al. introduced a dichotomic variable to quantum phase space and proposed a background Markov process for the time evolution of the Wigner function. However, in their method we need the whole distribution function to determine the next step of a particle under consideration. In this paper, we discuss a stationary quantum Markov process which enables us to treat each particle independently introducing U(1) extension for the phase space. The process has branching and vanishing points and we can keep a finite time interval between the branchings. The procedure to make the simulation of the process is also discussed.