Reverse engineer transition matrix from steady state?

Question

I had this open ended question. Is it possible to reverse engineer a transition matrix from the steady state. Is there a unique solution or could there be many? Basically given $P^{\infty}$ is there just one $P$ or many $P$'s and can they calculated?

Answer 1

There are always a lot of possible transition matrices for a given steady state, so many that I don't think it's useful to try to characterize them in general (though in a sense it's not that hard, since if you write out the stationarity condition it is still a linear problem in terms of the transition rates).

There are very useful formulas for calculating some transition rates, since this is one of the basic ingredients of Monte Carlo techniques in physics. A particular case of the Metropolis rule, for example:

Consider a system with $N$ states. Then

$$P_{xy} = \frac{\min (1,p_y/p_x)}{N-1}$$

for $x \ne y$ and

$$ P_{xx}=1-\sum_{y \ne x}P_{xy} $$ are the elements of a transition matrix with stationary state $p$.