A stochastic process, when subject to resetting to its initial condition at a
constant rate, generically reaches a non-equilibrium steady state. We study
analytically how the steady state is approached in time and find an unusual
relaxation mechanism in these systems. We show that as time progresses, an
inner core region around the resetting point reaches the steady state, while
the region outside the core is still transient. The boundaries of the core
region grow with time as power laws at late times. Alternatively, at a fixed
spatial point, the system undergoes a dynamical transition from the transient
to the steady state at a characteristic space dependent timescale t∗(x). We
calculate analytically in several examples the large deviation function
associated with this spatio-temporal fluctuation and show that generically it
has a second order discontinuity at a pair of critical points characterizing
the edges of the inner core. Our results are verified in the numerical
simulations of several models, such as simple diffusion and fluctuating
one-dimensional interfaces.Comment: 8 pages, 4 figures, published versio
