  A stochastic process with movement, return, and rest phases is considered in
this paper. For the movement phase, the particles move following the dynamics
of Gaussian process or ballistic type of L\'evy walk, and the time of each
movement is random. For the return phase, the particles will move back to the
origin with a constant velocity or acceleration or under the action of a
harmonic force after each movement, so that this phase can also be treated as a
non-instantaneous resetting. After each return, a rest with a random time at
the origin follows. The asymptotic behaviors of the mean squared displacements
with different kinds of movement dynamics, random resting time, and returning
are discussed. The stationary distributions are also considered when the
process is localized. Besides, the mean first passage time is considered when
the dynamic of movement phase is Brownian motion.

Statistical Mechanics