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 $\mathcal X(t)$ with stochastic resetting, starting from an initial position $\mathcal X(0)= \eta ;$ this type of diffusion $\mathcal X(t)$ is characterized by the fact that a reset to the position $x_R$ can occur according to a homogeneous Poisson process with rate $r>0.$ As regards the inverse first-passage place problem, for random $\eta \in (0,b), \ b < + \infty$ (and fixed $r$ and $x_R \in (0,b))$, let $\tau_{0,b}$ be the first time at which $\mathcal X(t)$ exits the interval $(0,b),$ and $\pi _0 = P(\mathcal X(\tau_{0,b}) = 0)$ the probability of exit from the left end of $(0,b);$ given a probability $q \in (0,1),$ the inverse first-passage place problem consists in finding the density $g$ of $\eta ,$ if it exists, such that $\pi _0 = q.$ Concerning the inverse first-passage time problem, for random $\eta \in (0, + \infty)$ (and fixed $r$ and $x_R >0)$, let $\tau$ be the first-passage time of $\mathcal X(t)$ 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 $\eta,$ if it exists, such that $P(\tau \le t ) = F(t), \ t >0.$ In addition to the case of random initial position $\eta,$ we also study the case when the initial position $\eta$ and the resetting rate $r$ are fixed, whereas the reset position $x_R$ is random. For all types of inverse problems considered, several explicit examples of solutions are reported.