Optimal conditions for first passage of jump processes with resetting

We investigate the first passage time beyond a barrier located at b≥0 of a random walk with independent and identically distributed jumps, starting from x0=0. The walk is subject to stochastic resetting, meaning that after each step the evolution is restarted with fixed probability r. We consider a resetting protocol that is an intermediate situation between a random walk (r=0) and an uncorrelated sequence of jumps all starting from the origin (r=1) and derive a general condition for determining when restarting the process with 0<r<1 is more efficient than restarting after each jump. If the mean first passage time of the process in the absence of resetting is larger than that of the sequence of jumps, this condition is sufficient to establish the existence of an optimal 0<r∗<1 that represents the best strategy, outperforming both r=0 and r=1. Our findings are discussed by considering two important examples of jump processes for which we draw the phase diagram illustrating the regions of the parameter space where resetting with some 0<r∗<1 is optimal.