Diffusion with doubly stochastic resetting

Diffusion with stochastic resetting, instantaneous returns of a diffusing particle to a reference point, creates a stationary probability distribution. The paradigm is extended here to a doubly stochastic protocol in which the resetting rate itself fluctuates in time and relaxes on its own timescale. An exact steady-state solution reveals three spatial regimes: a fluctuation-dominated core near the origin, a power-law regime at intermediate distances, and a far-field exponential decay fixed by the rate-relaxation time. These results expose how the instantaneous rate, mean rate, and relaxation time come together to determine the non-equilibrium steady state.