  The canonical Evans--Majumdar model for diffusion with stochastic resetting
to the origin assumes that resetting takes zero time: upon resetting the
diffusing particle is teleported back to the origin to start its motion anew.
However, in reality, getting from one place to another takes a finite amount of
time which must be accounted for as diffusion with resetting already serves as
a model for a myriad of processes in physics and beyond. Here we consider a
situation where upon resetting the diffusing particle returns to the origin at
a finite (rather than infinite) speed. This creates a coupling between the
particle's random position at the moment of resetting and its return time, and
further gives rise to a non-trivial cross-talk between two separate phases of
motion: the diffusive phase and the return phase. We show that each of these
phases relaxes to the stead-state in a unique manner; and while this could have
also rendered the total relaxation dynamics extremely non-trivial our analysis
surprisingly reveals otherwise. Indeed, the time-dependent distribution
describing the particle's position in our model is completely invariant to the
speed of return. Thus, whether returns are slow or fast, we always recover the
result originally obtained for diffusion with instantaneous returns to the
origin.
