Exit times from time-dependent random domains: continuity, weak convergence, and exit-time profiles Draft -currently under review at Stochastic Processes and their Applications

Abstract: 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.

• The paper introduces the non-tangency (NT) condition as a necessary and sufficient criterion for ensuring continuity of exit times from time-dependent domains.
• The paper demonstrates weak convergence of exit times using a continuous mapping theorem on scalarized barrier representations against joint perturbations.
• The paper extends the analysis to exit-time profiles via the Skorokhod M1 topology and provides practical verification methods in both jump and diffusion settings.

Exit Times from Time-Dependent Random Domains: Continuity, Weak Convergence, and Exit-Time Profiles

Introduction and Framework

The study addresses the continuity and convergence of exit times for stochastic processes from time-dependent domains, with emphasis on domains represented via continuous barrier functions on space-time. The classical approach—set convergence (such as Hausdorff or Fell topologies)—is demonstrated to be insufficient for guaranteeing convergence of exit/hitting times. Instead, the paper proposes a barrier/level-set representation $$\mathcal{T}_\phi = \{z \in \mathbb{R}^d : \phi(t, z) < 0\}$$, reducing the exit problem to a scalarized first-passage time problem. The central innovation is the identification of explicit conditions under which the exit-time functional is continuous with respect to joint perturbations of both the trajectory and the domain.

Deterministic Continuity and Non-Tangency (NT) Condition

A deterministic continuity theorem is provided in the Skorokhod local $J_1$ topology for paths and local uniform topology for barriers. The explicit non-tangency (NT) condition is introduced as necessary and sufficient for continuity—this condition encompasses configurations where the scalarized path strictly overshoots the barrier upon exit and rules out grazings or sticky behaviors at the boundary. The NT condition is proved to characterize the continuity set for the exit-time functional, substantiated by counter-examples showing that continuity fails when NT is violated, even under strong modes of convergence. The deterministic convergence theorem is strengthened by a careful analysis of potential discontinuity scenarios, particularly for paths that stick to the boundary.

Weak Convergence of Exit Times

The weak convergence result follows immediately from the deterministic continuity under NT via the continuous mapping theorem. This demonstrates that weak convergence of (path, barrier) pairs in the product Skorokhod-Barrier space guarantees weak convergence of exit times, provided NT is satisfied almost surely in the limit. Importantly, the result is pathwise and does not require independence or any structural assumptions between trajectory and domain, allowing application to a broad range of settings including random environments or feedback-coupled domains.

Functional Limit Theorem: Exit-Time Profiles and Skorokhod M1 Topology

The analysis extends to exit-time profiles indexed by barrier level $u$, i.e., the map $u \mapsto T(u)$, showing convergence in the Skorokhod $M_1$ topology. The $M_1$ topology is identified as essential for monotone, càdlàg first-passage profiles: $J_1$ is typically too strong due to misalignment in jump structure between approximating and limit exit-time profiles, as illustrated by explicit examples. The convergence theory for functional exit-time profiles hinges on regularity of rational levels for the limit scalarized path, leveraging the monotonicity and càdlàg structure of the profile to reduce the topological requirements.

Verification Routes for Non-Tangency and Practical Example

Two principal verification routes for NT are provided:

• Jump Overshoot Condition: Directly applicable when crossing occurs via jumps, ensuring immediate separation from the boundary after crossing.
• Continuous Semimartingale/Itô Non-Characteristic Criterion: In diffusion settings, verified by non-degeneracy of the martingale component in the boundary-normal direction, ensuring robust local overshoot.

A worked example demonstrates the full framework: for a random walk converging to Brownian motion and a smooth moving boundary, joint convergence is easily established, NT verified via Itô calculus, and both exit-time and exit-time profile convergence results in $M_1$ topology are obtained.

Implications and Future Directions

The results provide a robust foundation for weak convergence of exit times in stochastic processes with time-dependent or random domains, with precise boundary delineation given by NT. This provides a unifying framework amenable to generalization and critical for numerical approximations in level-set methods and viscosity-solution schemes. As practical implications, the continuity and weak convergence results enable principled usage of limiting arguments, discretization schemes, and stochastic numerics involving dynamic and evolving boundaries.

Theoretically, the identification of NT as the sharp continuity criterion sets the stage for further research into more general domain representations, incorporating barriers with time-discontinuities or domain-state coupling. There is potential for extension to high-dimensional or infinite-dimensional settings, stochastic PDEs with moving boundaries, and models involving endogenous boundary evolution (e.g., optimal stopping or free boundary problems).

Conclusion

The paper rigorously establishes deterministic continuity and weak convergence results for exit times from time-dependent random domains under joint convergence of path and barrier, characterized by the non-tangency condition. The development elucidates the necessity of barrier representations and the Skorokhod $J_1$ and $M_1$ topologies, and provides verification routes applicable in classical diffusion and jump-overshoot scenarios. The framework is instrumental for both theoretical limit theorems and practical stochastic simulation in evolving domains, and invites further exploration of boundary-crossing phenomena in more general stochastic systems.