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A stochastic process, when subject to resetting to its initial condition at aconstant rate, generically reaches a non-equilibrium steady state. We studyanalytically how the steady state is approached in time and find an unusualrelaxation mechanism in these systems. We show that as time progresses, aninner core region around the resetting point reaches the steady state, whilethe region outside the core is still transient. The boundaries of the coreregion grow with time as power laws at late times. Alternatively, at a fixedspatial point, the system undergoes a dynamical transition from the transientto the steady state at a characteristic space dependent timescale $t^*(x)$. Wecalculate analytically in several examples the large deviation functionassociated with this spatio-temporal fluctuation and show that generically ithas a second order discontinuity at a pair of critical points characterizingthe edges of the inner core. Our results are verified in the numericalsimulations of several models, such as simple diffusion and fluctuatingone-dimensional interfaces.

Dynamical transition in the temporal relaxation of stochastic processes under resetting