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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2604.03129 |
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| _version_ | 1866908934990725120 |
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| author | Guillaume, Tristan |
| author_facet | Guillaume, Tristan |
| contents | We study exit times from time-dependent domains under joint perturbations of the trajectory and the domain. Representing a moving domain by a continuous barrier $Φ$ on space-time, we reduce the exit problem to a one-dimensional first-passage problem for the scalarised path $y(t) := Φ(t,x(t))$. Our first main result is a deterministic continuity theorem: the exit-time functional is continuous, under local Skorokhod $J_1$ convergence of the path and local uniform convergence of the barrier, at every configuration satisfying an explicit non-tangency condition (NT). We show that NT is sharp in the sense that it characterises the continuity set of the functional. As a direct consequence, weak convergence of exit times follows from joint weak convergence of paths and barriers whenever the limiting pair satisfies NT almost surely; no independence or structural restrictions between trajectory and domain are required. Our second main result is a functional limit theorem: the exit-time profile $u\mapstoτ(u)$, viewed as a càdlàg function of the barrier level, converges in the Skorokhod $M_1$ topology under the same hypotheses, with a concrete example showing that $J_1$ convergence can fail. Concrete verification routes for NT are provided, including a non-characteristic/Itô criterion for diffusions, and the full framework is illustrated through a worked Donsker-type example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03129 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exit times from time-dependent random domains: continuity, weak convergence, and exit-time profiles Draft -currently under review at Stochastic Processes and their Applications Guillaume, Tristan Probability We study exit times from time-dependent domains under joint perturbations of the trajectory and the domain. Representing a moving domain by a continuous barrier $Φ$ on space-time, we reduce the exit problem to a one-dimensional first-passage problem for the scalarised path $y(t) := Φ(t,x(t))$. Our first main result is a deterministic continuity theorem: the exit-time functional is continuous, under local Skorokhod $J_1$ convergence of the path and local uniform convergence of the barrier, at every configuration satisfying an explicit non-tangency condition (NT). We show that NT is sharp in the sense that it characterises the continuity set of the functional. As a direct consequence, weak convergence of exit times follows from joint weak convergence of paths and barriers whenever the limiting pair satisfies NT almost surely; no independence or structural restrictions between trajectory and domain are required. Our second main result is a functional limit theorem: the exit-time profile $u\mapstoτ(u)$, viewed as a càdlàg function of the barrier level, converges in the Skorokhod $M_1$ topology under the same hypotheses, with a concrete example showing that $J_1$ convergence can fail. Concrete verification routes for NT are provided, including a non-characteristic/Itô criterion for diffusions, and the full framework is illustrated through a worked Donsker-type example. |
| title | Exit times from time-dependent random domains: continuity, weak convergence, and exit-time profiles Draft -currently under review at Stochastic Processes and their Applications |
| topic | Probability |
| url | https://arxiv.org/abs/2604.03129 |