Study of direct and inverse first-exit problems for drifted Brownian motion with Poissonian resetting

Abstract

\noindent We address some direct and inverse problems, for the first-exit time (FET) $\tau$ of a drifted Brownian motion with Poissonian resetting ${\cal X}(t)$ from an interval $(0,b)$ and the first-exit area (FEA) $A,$ namely the area swept out by ${\cal X}(t)$ till the time $\tau$; this type of diffusion process ${\cal X}(t)$ is characterized by the fact that a reset to the position $x_R$ can occur according to a homogeneous Poisson process with rate $r>0.$ When the initial position ${\cal X}(0)= \eta \in (0,b)$ is deterministic and fixed, the direct FET problem consists in investigating the statistical properties of the FET $\tau ,$ whilst the direct FEA problem studies the probability distribution of the FEA $A$. The inverse FET problem regards the case when $\eta$ is randomly distributed in $(0,b)$ (while $r$ and $x_R$ are fixed); if $F(t)$ is a given distribution function on the time $t$ axis, the inverse FET 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. Several explicit examples of solutions to the inverse FET problem are provided.