Resetting processes with noninstantaneous return

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 2020. Vol. 101. No. 5. P. 052130.

Bodrova A., Sokolov I. M.

We consider a random two-phase process which we call a reset-return one. The particle starts
its motion at the origin. The rst, displacement, phase corresponds to a stochastic motion of a
particle and is nished 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 di erent 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 xed time. For the rst
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.

Keywords: anomalous diffusion, аномальная диффузия