Steady-state solution of Markov jump processes in terms of arrival probabilities

Several dynamical processes can be modeled as Markov jump processes among a finite number N of sites (the distinct physical states). Here we consider strongly connected networks with time-independent site-to-site jump rate constants, and focus on the steady-state occupation probabilities of the sites. We provide a physically framed expression of the steady-state distribution in terms of arrival probabilities, here defined as the probabilities of going from starting sites to target sites with a given number of jumps (regardless of the time required). In particular, the full set of return probabilities (for all the sites of the network) up to N-1 jumps is necessary and sufficient. A few examples illustrate the outcomes, including the case of stochastic chemical kinetics.