Inverse First-Passage Problems of a Diffusion with Resetting

We address some inverse problems for the first-passage place and the first-passage time of a one-dimensional diffusion process with stochastic resetting. This type of diffusion is characterized by the fact that a reset to the position xR can occur according to a homogeneous Poisson process with rate r > 0. As regards the inverse first-passage place problem, for random initial position belonging to an interval (0, b) with finite b > 0 (and fixed r and xR belonging to (0, b)), let τ be the first time at which the process exits the interval (0, b), and π0 be the probability of exit from the left end of (0, b). Given a probability q, the inverse first-passage place problem consists in finding the density g of the initial position, if it exists, such that π0 = q. Concerning the inverse first-passage time problem, for random positive starting point (and fixed r and xR > 0), let again τ be the first-passage time of the process through zero. For a given distribution function F(t) on the positive real axis, the inverse first-passage time problem consists in finding the density g of the starting point, if it exists, such that P(τ ≤ t) = F(t), t > 0. In addition to the case of random initial position, we also study the case when the initial position and the resetting rate r are fixed, whereas the reset position xR is random. For all types of inverse problems considered, several explicit examples of solutions are reported.