Non-i.i.d. random holomorphic dynamical systems and the probability of tending to infinity

We consider random holomorphic dynamical systems on the Riemann sphere whose choices of maps are related to Markov chains. Our motivation is to generalize the facts which hold in i.i.d. random holomorphic dynamical systems. In particular, we focus on the function which represents the probability of tending to infinity. We show some sufficient conditions which make continuous on the whole space and we characterize the Julia sets in terms of the function under certain assumptions.