Abstract
We consider the problem of the first passage time to the origin of a spatially non-homogeneous random walk (RW) with a position-dependent drift, known as the Gillis random walk, in the presence of resetting. The walk starts from an initial site x (0) and, with fixed probability r, at each step may be relocated to a given site x ( r ). From a general perspective, we first derive a series of results regarding the first and the second moment of the first hitting time distribution, valid for a wide class of processes, including RWs lacking the property of translational invariance; we then apply these results to the specific model. When resetting is not applied, by tuning the value of a parameter which defines the transition probability of the process, denoted by epsilon, the recurrence properties of the walk are changed, and we can observe: a transient walk, a null-recurrent walk, or a positive-recurrent walk. When the resetting mechanism is switched on, we study quantitatively in all regimes the improvement of the search efficiency. In particular, in every case resetting allows the system to reach the target with probability one and, on average, in a finite time. If the reset-free system is in the transient or null-recurrent regime, this makes resetting always advantageous and moreover, it assures the existence of an optimal resetting probability r*
