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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2502.19901 |
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| _version_ | 1866929734135316480 |
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| author | Abundo, Mario |
| author_facet | Abundo, Mario |
| contents | \noindent We address some direct and inverse problems, for the first-exit time (FET) $τ$ 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 $τ$; 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 ${\cal X}(0)= η\in (0,b)$ is deterministic and fixed, the direct FET problem consists in investigating the statistical properties of the FET $τ,$ whilst the direct FEA problem studies the probability distribution of the FEA $A$. The inverse FET problem regards the case when $η$ 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 $η,$ if it exists, such that $P[τ\le t ] = F(t), \ t >0.$ %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 $x_R$ is random. Several explicit examples of solutions to the inverse FET problem are provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19901 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Study of direct and inverse first-exit problems for drifted Brownian motion with Poissonian resetting Abundo, Mario Probability 60J60, 60H05, 60H10 \noindent We address some direct and inverse problems, for the first-exit time (FET) $τ$ 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 $τ$; 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 ${\cal X}(0)= η\in (0,b)$ is deterministic and fixed, the direct FET problem consists in investigating the statistical properties of the FET $τ,$ whilst the direct FEA problem studies the probability distribution of the FEA $A$. The inverse FET problem regards the case when $η$ 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 $η,$ if it exists, such that $P[τ\le t ] = F(t), \ t >0.$ %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 $x_R$ is random. Several explicit examples of solutions to the inverse FET problem are provided. |
| title | Study of direct and inverse first-exit problems for drifted Brownian motion with Poissonian resetting |
| topic | Probability 60J60, 60H05, 60H10 |
| url | https://arxiv.org/abs/2502.19901 |