pith. machine review for the scientific record. sign in

arxiv: 2604.03129 · v1 · submitted 2026-04-03 · 🧮 math.PR

Recognition: 2 theorem links

· Lean Theorem

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

Tristan Guillaume (CYU)
Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:37 UTC · model grok-4.3

classification 🧮 math.PR
keywords exit timestime-dependent domainsSkorokhod topologynon-tangency conditionweak convergencefunctional limit theoremfirst-passage times
0
0 comments X

The pith

The exit-time functional is continuous under local Skorokhod J1 path convergence and local uniform barrier convergence exactly when the non-tangency condition NT holds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reduces exit from a time-dependent domain to a one-dimensional first-passage problem by scalarising the path against the barrier function. It proves that this exit-time map stays continuous at every pair where the path does not graze the barrier tangentially at the exit instant, and shows that this non-tangency condition fully characterises the continuity set. Because the result is deterministic, joint weak convergence of paths and barriers immediately yields weak convergence of exit times whenever the limit satisfies NT almost surely, with no independence or other structural assumptions required. The same hypotheses also give convergence of the entire exit-time profile viewed as a function of barrier level, but only in the M1 topology.

Core claim

We prove a deterministic continuity theorem for the exit-time functional under local Skorokhod J1 convergence of the path and local uniform convergence of the barrier at every configuration satisfying the explicit non-tangency condition NT. NT is sharp because 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. The exit-time profile converges in the Skorokhod M1 topology under the same hypotheses, although J1 convergence can fail.

What carries the argument

The non-tangency condition NT, which requires that the scalarised path does not touch the barrier tangentially at the exit time and thereby guarantees that small perturbations do not produce large jumps in the exit time.

If this is right

  • Weak convergence of exit times is inherited directly from joint weak convergence of paths and barriers under the NT condition.
  • The exit-time profile as a function of barrier level converges in the Skorokhod M1 topology.
  • Verification of NT is possible through an explicit non-characteristic or Itô-type criterion for diffusions.
  • The results apply to Donsker-type approximations without any independence requirement between trajectory and domain.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reduction to a scalar first-passage problem may allow similar continuity arguments to be applied to other moving-boundary problems in stochastic processes.
  • In applications such as time-dependent barrier options or reliability models, the theorems justify approximating the domain and trajectory separately provided tangential contact has probability zero in the limit.
  • Numerical simulation of specific diffusions could test whether NT holds with high probability for given families of moving barriers.

Load-bearing premise

The limiting path and barrier pair satisfies the non-tangency condition almost surely.

What would settle it

A sequence of paths and barriers that converge jointly in the local Skorokhod J1 and local uniform topologies on a set of positive probability where NT fails, yet the corresponding exit times fail to converge to the exit time of the limit.

read the original 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 $\Phi$ on space-time, we reduce the exit problem to a one-dimensional first-passage problem for the scalarised path $y(t) := \Phi(t,x(t))$. 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 $u\mapsto\tau(u)$, viewed as a c\`adl\`ag 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/It\^o criterion for diffusions, and the full framework is illustrated through a worked Donsker-type example.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies exit times from time-dependent domains represented by continuous barriers Φ. It reduces the problem to a scalar first-passage time for y(t) = Φ(t, x(t)), proves a deterministic continuity theorem for the exit-time functional under local Skorokhod J1 path convergence and local uniform barrier convergence at configurations satisfying an explicit non-tangency condition (NT), shows NT characterizes the continuity set, derives weak convergence of exit times from joint weak convergence of paths and barriers whenever the limit satisfies NT a.s. (no independence required), and establishes M1 convergence of the exit-time profile u ↦ τ(u) with an explicit counter-example that J1 can fail. Concrete verification routes for NT are supplied, including an Itô/non-characteristic criterion for diffusions and a Donsker-type example.

Significance. If the results hold, the work supplies a general, assumption-light framework for continuity and weak convergence of exit times under joint perturbations of paths and domains. The deterministic character of the continuity theorem, the sharpness of NT as the exact continuity set, the absence of independence requirements, and the M1 functional limit (with J1 counter-example) are genuine strengths. The supplied verification routes (non-characteristic/Itô criterion and Donsker illustration) make the results usable in applications.

minor comments (3)
  1. The abstract and introduction should explicitly state the precise definition of the non-tangency condition (NT) rather than deferring it entirely to a later section, to improve readability for readers focused on the main theorems.
  2. In the statement of the functional limit theorem, clarify whether the M1 topology is taken with respect to the uniform metric on the barrier levels or the Skorokhod metric on the profile; a short remark on the choice would prevent ambiguity.
  3. The Donsker-type example would benefit from an explicit statement of the scaling regime and the limiting process to make the illustration self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript. The report correctly identifies the deterministic continuity theorem under local Skorokhod J1 path convergence and local uniform barrier convergence at non-tangency points, the characterization of the continuity set, the weak convergence result without independence assumptions, and the M1 functional convergence of the exit-time profile together with the explicit J1 counter-example. We also appreciate the recognition of the supplied verification routes (Itô/non-characteristic criterion and Donsker illustration). No specific major comments were raised in the report. We will therefore implement only minor editorial polishing and clarification of notation in the revised version.

Circularity Check

0 steps flagged

No significant circularity: deterministic topological argument is self-contained

full rationale

The derivation reduces the exit-time problem to the scalar first-passage time of y(t) := Φ(t, x(t)), then establishes continuity of the exit-time functional under local J1 path convergence and local uniform barrier convergence precisely on the set where the explicit non-tangency condition (NT) holds. NT is shown to characterize the continuity set by direct construction of counter-examples when it fails. Weak convergence and the M1 functional limit then follow immediately from the deterministic continuity result whenever the limiting pair satisfies NT almost surely. No parameter is fitted to data, no prediction is obtained by renaming a fitted quantity, and no load-bearing step relies on a self-citation whose content is itself unverified. The argument is purely topological and measure-theoretic; all steps are externally verifiable from the stated hypotheses without circular reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on standard properties of Skorokhod topologies and weak convergence; the non-tangency condition is introduced as the key new assumption that characterizes continuity.

axioms (2)
  • standard math Properties of the Skorokhod J1 and M1 topologies on cadlag functions
    Invoked for convergence of paths and barrier functions.
  • standard math Standard weak convergence results in metric spaces
    Used to transfer convergence from paths and barriers to exit times.
invented entities (1)
  • Non-tangency condition (NT) no independent evidence
    purpose: Characterizes the set of continuity points of the exit-time functional
    Explicitly defined to exclude tangential exits; no independent evidence outside the paper is provided.

pith-pipeline@v0.9.0 · 5565 in / 1448 out tokens · 65842 ms · 2026-05-13T18:37:07.303330+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Barles, P.E

    G. Barles, P.E. Souganidis, Convergence of approximati on schemes for fully nonlinear second order equations, Asymptotic Anal. 4 (1991) 271–

    work page 1991

  2. [2]

    https://doi.org/10.3233/ASY-1991-4305

    work page doi:10.3233/asy-1991-4305 1991

  3. [3]

    Billingsley, Convergence of Probability Measures, 2 nd ed., Wiley, New York, 1999

    P. Billingsley, Convergence of Probability Measures, 2 nd ed., Wiley, New York, 1999

    work page 1999

  4. [4]

    C. Cai, T. De Angelis, and J. Palczewski, On the Continuit y of Optimal Stopping Surfaces for Jump-Diffusions, SIAM J. Control Opti m. 61(3) (2023) 1513–1531. doi:10.1137/21M1448094

    work page doi:10.1137/21m1448094 2023

  5. [5]

    Crandall, H

    M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to vis cosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992) 1–67. https://doi.org/10.1090/S0273-0979-1992- 00266-5

    work page doi:10.1090/s0273-0979-1992- 1992

  6. [6]

    Fell, A Hausdorff topology for the closed subsets o f a locally compact non-Hausdorff space, Proc

    J.M.G. Fell, A Hausdorff topology for the closed subsets o f a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962 ) 472– 476: 10.1090/S0002-9939-1962-0139135-6 28

    work page doi:10.1090/s0002-9939-1962-0139135-6 1962

  7. [7]

    Gibou, R

    F. Gibou, R. Fedkiw, S. Osher, A review of level-set metho ds and some recent applications, J. Comput. Phys. 353 (2018) 82 –109. https://doi.org/10.1016/j.jcp.2017.10.024

    work page doi:10.1016/j.jcp.2017.10.024 2018

  8. [8]

    Jacod, A.N

    J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic P rocesses, 2nd ed., Springer, Berlin, 2003

    work page 2003

  9. [9]

    Liang, K

    V. Liang, K. Borovkov, On time-dependent boundary cross - ing probabilities of diffusion processes as differentiable f unction- als of the boundary, Stoch. Proc. Appl. 190 (2025) 104742. https://doi.org/10.1016/j.spa.2025.104742

    work page doi:10.1016/j.spa.2025.104742 2025

  10. [10]

    Matheron, Random Sets and Integral Geometry, Wiley, N ew York, 1975

    G. Matheron, Random Sets and Integral Geometry, Wiley, N ew York, 1975

    work page 1975

  11. [11]

    Molchanov, Theory of Random Sets, Springer, London, 2005

    I. Molchanov, Theory of Random Sets, Springer, London, 2005. https://doi.org/10.1007/1-84628-150-4

    work page doi:10.1007/1-84628-150-4 2005

  12. [12]

    Nyström, T

    K. Nyström, T. Önskog, The Skorokhod oblique reflection problem in time-dependent domains, Ann. Probab. 38 (2010) 2170–222 3. DOI: 10.1214/10-AOP538

    work page doi:10.1214/10-aop538 2010

  13. [13]

    Osher, J.A

    S. Osher, J.A. Sethian, Fronts propagating with curvat ure- dependent speed: algorithms based on Hamilton– Jacobi formulations, J. Comput. Phys. 79 (1988) 12–49. https://doi.org/10.1016/0021-9991(88)90002-2

    work page doi:10.1016/0021-9991(88)90002-2 1988

  14. [14]

    Peskir, A change-of-variable formula with local tim e on curves, J

    G. Peskir, A change-of-variable formula with local tim e on curves, J. Theoret. Probab. 18 (2005) 499–535. DOI: 10.1007/s10959-0 05-3517-6

    work page doi:10.1007/s10959-0 2005

  15. [15]

    Puhalskii, W

    A.A. Puhalskii, W. Whitt, Functional large deviation p rinciples for first- passage-time processes, Ann. Appl. Probab. 7 (1997) 362–38 1. DOI: 10.1214/aoap/1034625336

    work page doi:10.1214/aoap/1034625336 1997

  16. [16]

    Limit theorems for stochastic proces ses,

    A.V. Skorokhod, "Limit theorems for stochastic proces ses," Theory Probab. Appl. 1 (1956) 261–290: 10.1137/1101022

    work page doi:10.1137/1101022 1956

  17. [17]

    Soner, V

    H.M. Soner, V. Tissot-Daguette, Stopping times of boun daries: relax- ation and continuity, SIAM J. Control Optim. 63 (2025) 2835– 2855. https://doi.org/10.1137/23M1596761. 29

    work page doi:10.1137/23m1596761 2025

  18. [18]

    Vervaat, Functional central limit theorems for proc esses with positive drift and their inverses, Z

    W. Vervaat, Functional central limit theorems for proc esses with positive drift and their inverses, Z. Wahrscheinlichkeitstheorie v erw. Gebiete 23 (1972) 245–253: 10.1007/BF00532510

    work page doi:10.1007/bf00532510 1972

  19. [19]

    Whitt, Weak convergence of first-passage time proces ses, J

    W. Whitt, Weak convergence of first-passage time proces ses, J. Appl. Probab. 8 (1971) 417–422: 10.2307/3211913

    work page doi:10.2307/3211913 1971

  20. [20]

    Whitt, Some useful functions for functional limit th eorems, Math

    W. Whitt, Some useful functions for functional limit th eorems, Math. Oper. Res. 5 (1980) 67–85. DOI 10.1287/moor.5.1.67

    work page doi:10.1287/moor.5.1.67 1980

  21. [21]

    Whitt, Stochastic-Process Limits: An Introduction to Stochastic- Process Limits and Their Application to Queues, Springer, N ew York, 2002

    W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic- Process Limits and Their Application to Queues, Springer, N ew York, 2002. Appendix A. Proof of Lemma 4.4 Lemma 4.4 (restated) Let a < b . Let fn, f ∈ D([a, b ], R) be nondecreasing and càdlàg. Suppose there is a dense set D0 ⊂ [a, b ] containing a, b such that fn(t) → f (t) for all t ∈ D...

    work page 2002