In the standard treatment of a markov process, the state vector is a probability vector, whose elements can be between zero and one. But I have a need to constrain the state vector to zeros or ones. To be more precise, for my work, I need to constrain a state vector to be all zeros, and a single one, so that at every point in time the system is in a specific state. I think this may be referred to as an "absolute state vector".
To give the system a history, with absolute state vectors, I must sample the transition matrix at each point in time in such a way that it consists of only zeros and ones. If this is done properly, then the system moves from one absolute state to another. So the system stays in a definite state at every time.
I know that it is possible to treat a markov process this way, but I can't find anywhere that somebody has done it. I don't want to reinvent terminology or notation. Can anyone please point me in the direction of where this treatment of a markov process has been established?
Thanks.