  In the past few years, stochastic resetting has become a subject of immense
interest. Most of the theoretical studies so far focused on instantaneous
resetting which is, however, a major impediment to practical realization or
experimental verification in the field. This is because in the real world,
taking a particle from one place to another requires finite time and thus a
generalization of the existing theory to incorporate non-instantaneous
resetting is very much in need. In this paper, we propose a method of resetting
which involves non-instantaneous returns facilitated by an external confining
trap potential $U(x)$ centered at the resetting location. We consider a
Brownian particle that starts its random motion from the origin. Upon
resetting, the trap is switched on and the particle starts experiencing a force
towards the center of the trap which drives it to return to the origin. The
return phase ends when the particle makes a first passage to this center. We
develop a general framework to study such a set up. Importantly, we observe
that the system reaches a non-equilibrium steady state which we analyze in full
details for two choices of $U(x)$, namely, (i) linear and (ii) harmonic.
Finally, we perform numerical simulations and find an excellent agreement with
the theory. The general formalism developed here can be applied to more
realistic return protocols opening up a panorama of possibilities for further
theoretical and experimental applications.
