  We consider a random two-phase process which we call a reset-return one. The
particle starts its motion at the origin. The first, displacement, phase
corresponds to a stochastic motion of a particle and is finished at a resetting
event. The second, return, phase corresponds to the particle's motion towards
the origin from the position it attained at the end of the displacement phase.
This motion towards the origin takes place according to a given equation of
motion. The whole process is a renewal one. We provide general expressions for
the stationary probability density function of the particle's position and for
the mean hitting time in one dimension. We perform explicit analysis for the
Brownian motion during the displacement phase and three different types of the
return motion: return at a constant speed, return at a constant acceleration
with zero initial speed and return under the action of a harmonic force. We
assume that the waiting times for resetting events follow an exponential
distribution, or that resetting takes place at a constant pace. For the first
two types of return motion and the exponential resetting the stationary
probability density function of the particle's position is invariant under
return speed (acceleration), while no such invariance is found for
deterministic resetting, and for exponential resetting with return under the
action of the harmonic force. We discuss necessary conditions for such
invariance of the stationary PDF of the positions with respect to the
properties of the return process, and demonstrate some additional examples when
this invariance does or does not take place.
