Optimization in the first-passage problem of a diffusion with Poissonian resetting

We address the problem of minimizing the expected first-passage time of a Brownian motion with Poissonian resetting, with respect to the resetting rate $r.$ We consider both the one-boundary and the two-boundary cases.We investigate the first-passage time (FPT) and first-exit time (FET) of a one-dimensional, time-homogeneous diffusion process subject to Poissonian resetting. We first derive a general analytical relationship that expresses the Laplace transform (LT) and the expected value of the FPT (and FET) for the process with resetting in terms of the LT of the FPT (and FET) of the underlying diffusion without resetting. This framework is then applied to determine the optimal resetting rate $r$ that minimizes the expected FPT (and FET). We provide explicit results for drifted Brownian motion and Ornstein-Uhlenbeck (OU) process. For Brownian motion, we extend existing literature by considering the case where the initial position $x$ differs from the resetting position $x_R$, providing a comprehensive parametric analysis. For the OU process, we provide new insights into the minimization of the expected FPT. Our results demonstrate how a strategic choice of the resetting rate can effectively regularize and accelerate the passage through one or two boundaries.