2024/10/22 by Mario Abundo, Abundo, Mario
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2410.16889
openalex publication_date 2024/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)= η; this type of diffusion \mathcal X(t) 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 η∈ (0,b), b < + ∞ (and fixed r and xR ∈ (0,b)), let τ0,b be the first time at which \mathcal X(t) exits the interval (0,b), and π0 = P(\mathcal X(τ0,b) = 0) the probability of exit from the left end of (0,b); given a probability q ∈ (0,1), the inverse first-passage place problem consists in finding the density g of η, if it exists, such that π0 = q. Concerning the inverse first-passage time problem, for random η∈ (0, + ∞) (and fixed r and xR >0), let τ 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 η, if it exists, such that P(τ≤ t ) = F(t), t >0. In addition to the case of random initial position η, we also study the case when the initial position η 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.