Occupation Times in Markov Processes

We consider, in a homogeneous Markov process with finite state space, the occupation times that is, the times spent by the process in given subsets of the state space during a finite interval of time. We first derive the distribution of the occupation time of one subset and then we generalize this result to the joint distribution of occupation times of different subsets of the state space by the use of order statistics from the uniform distribution. Next, we consider the distribution of weighted sums of occupation times. We obtain the forward and backward equations describing the behavior of these weighted sums and we show how these equations lead to simple expressions of this distribution