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We consider motion of an overdamped Brownian particle subject to stochasticresetting in one dimension. In contrast to the usual setting where the particleis instantaneously reset to a preferred location (say, the origin), here weconsider a finite time resetting process facilitated by an external linearpotential $V(x)=\lambda|x|~ (\lambda>0)$. When resetting occurs, the trap isswitched on and the particle experiences a force $-\partial_x V(x)$ which helpsthe particle to return to the resetting location. The trap is switched off assoon as the particle makes a first passage to the origin. Subsequently, theparticle resumes its free diffusion motion and the process keeps repeating. Inthis set-up, the system attains a non-equilibrium steady state. We study therelaxation to this steady state by analytically computing the positiondistribution of the particle at all time and then analysing this distributionusing the spectral properties of the corresponding Fokker-Planck operator. Asseen for the instantaneous resetting problem, we observe a `cone spreading'relaxation with travelling fronts such that there is an inner core regionaround the resetting point that reaches the steady state, while the regionoutside the core still grows ballistically with time. In addition to theunusual relaxation phenomena, we compute the large deviation functionsassociated to the corresponding probability density and find that the largedeviation functions describe a dynamical transition similar to what is seenpreviously in case of instantaneous resetting. Notably, our method, based onspectral properties, complements the existing renewal formalism and reveals theintricate mathematical structure responsible for such relaxation phenomena. Weverify our analytical results against extensive numerical simulations.

Resetting with stochastic return through linear confining potential