First-passage time in space-dependent stochastic resetting

We consider the mean first-passage time (MFPT) through a target of interest for a diffusive particle of Langevin type, with the added condition that the particle is reset to its original position with some rate r. We study both smooth and nonsmooth, nonconvex potentials, focusing on the case where the reset rate depends on the space coordinate. For quadratic and piecewise-quadratic potentials, we show that the benefits of resetting depend on the ratio between drift and noise, and become more important as the drift potential becomes smaller compared to the noise. When the target is a local optimum of the potential, we further show that it is beneficial to use a space-dependent resetting where the reset rate is lower when the particle is closer to the target.