Jump statistics, sojourn times, fluctuation dynamics and ergodic behavior for Markov processes in continuous time with a finite number of states

A general approach is introduced for describing the time evolution of a Markov process in continuous time and with a finite number of states. The total number of transition events from one state to other states and of the total sojourn times of the system in the different states are used as additional state variables. The large time behavior of these two types of stochastic state variables is investigated analytically by using a stochastic Liouville equation. It is shown that the cumulants of first and second order of the state variables increase asymptotically linearly in time. A set of scaled sojourn times is introduced which in the limit of large times have a Gaussian behavior. For long times, the total average sojourn times are proportional to the stationary state probability of the process and, even though the relative fluctuations decrease to zero, the relative cross correlation functions tend towards finite values. The results are used for investigating the connections with Van Kampenʹs approach for investigating the ergodic properties of Markov processes. The theory may be applied for studying fluctuation dynamics in stochastic reaction diffusion systems and for computing effective rates and transport coefficients for non-equilibrium processes in systems with dynamical disorder.