Renewal theory for a run-and-tumble particle with stochastic resetting and a sticky boundary

In this paper, we consider a run-and-tumble particle (RTP) with stochastic resetting confined to the half line [0,∞) with a sticky boundary at x=0. In the bulk, the RTP tumbles at a constant rate α>0 between velocity states ±v with v>0 and randomly resets to its initial position and orientation (x_{0},k_{0})∈(R^{+},±). When the RTP reaches the target at x=0, it attaches to the boundary for some random waiting time before either detaching and continuing to navigate the bulk domain or (permanently) entering the target. These events are the analogs of adsorption, desorption, and absorption of a particle by a partially reactive surface in physical chemistry. We use renewal theory to characterize the particle trajectory in terms of successive binding events at x=0 under two distinct desorption protocols: via resetting to (x_{0},k_{0}) and via continuous movement from x=0 with velocity +v. First, we derive the nonequilibrium stationary state (NESS) in the case of no absorption and characterize the accumulation at the boundary. Second, we compute the mean first passage time (MFPT) statistics. In addition to observing the usual unimodal dependence of the MFPT on bulk resetting, both the NESS and MFPT strongly depend on the initial orientation k_{0} and the desorption protocol. For instance, if the initial orientation is toward the boundary, we find that the desorption-induced resetting protocol can reduce the MFPT more effectively than the nonresetting desorption protocol. We also show how matching the desorption kinetics with the bulk resetting or tumbling rate introduces a tradeoff between minimizing the adsorption and absorption times. In this setting, we find that the desorption protocol which minimizes the absorption MFPT for a given set of parameters is almost always the opposite of that favored when desorption and bulk kinetics are not the same. These results collectively highlight the utility of the renewal formalism in characterizing two distinct desorption protocols particular to an RTP.