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

Aging two-state process with L\'{e}vy walk and Brownian motion