  With the rich dynamics studies of single-state processes, the two-state
processes attract more and more interests of people, since they are widely
observed in complex system and have effective applications in diverse fields,
say, foraging behavior of animals. This report builds the theoretical
foundation of the process with two states: L\'{e}vy walk and Brownian motion,
having been proved to be an efficient intermittent search process. The sojourn
time distributions in two states are both assumed to be heavy-tailed with
exponents $\alpha_\pm\in(0,2)$. The dynamical behaviors of this two-state
process are obtained through analyzing the ensemble-averaged and time-averaged
mean squared displacements (MSDs) in weak and strong aging cases. It is
discovered that the magnitude relationship of $\alpha_\pm$ decides the fraction
of two states for long times, playing a crucial role in these MSDs. According
to the generic expressions of MSDs, some inherent characteristics of the
two-state process are detected. The effects of the fraction on these
observables are detailedly presented in six different cases. The key of getting
these results is to calculate the velocity correlation function of the
two-state process, the techniques of which can be generalized to other
multi-state processes.
