Stochastic movement subject to a reset-and-residence mechanism: transport properties and first arrival statistics

In this work, we consider stochastic movement with random resets to the origin followed by a random residence time before the motion starts again. First, we study the transport properties of the walker, i.e. we derive an expression for the mean square displacement of the overall process and study its dependence on the statistical properties of the resets and the residence times probability density functions (PDFs) and the type of movement. From this general formula, we see that the inclusion of the residence after the resets is able to induce super-diffusive to sub-diffusive (or diffusive) regimes and it can also make a sub-diffusive walker reach a constant mean square displacement or even collapse. Second, we study how the reset-and-residence mechanism affects the survival probability of different search processes to a given position, showing that the long time behavior of the reset and residence time PDFs determine the existence of the mean first arrival time.