Diffusion in Heterogeneous Media with Stochastic Resetting and Pauses

Diffusion in heterogeneous environments is usually governed by unusual dynamics, exhibiting sub- or superdiffusive scaling depending on the structural complexity and memory effects. In many systems, diffusing particles may alternate between periods of motion and rest, or may undergo stochastic resetting to a preferred position. While each of these mechanisms has been studied independently, their combined effect in a heterogeneous medium has been insufficiently investigated. We formulate and solve a coupled set of one dimension diffusion equations for the probability densities of moving and resting particles, accounting for space-dependent diffusivity and stochastic resetting. We obtain expressions for the probability distribution and show the behavior of the survival probability, mean-square displacement, and first-passage time. The results reveal a diverse range of behaviors with distinct diffusion regimes. One of them is obtained for small times, which can be connected to the heterogeneity present in the system, and another for intermediate times related to the intermittent process produced by the moving and pauses before the system reaches the stationary state.