Resetting processes with non-instanteneous return

Abstract: 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.