pith. sign in

arxiv: 2601.11445 · v3 · submitted 2026-01-16 · 🧮 math.PR · math.OC

Stochastic Perturbation of Sweeping Processes Driven by Continuous Uniformly Prox-Regular Moving Sets

Juan Guillermo Garrido , Nabil Kazi-Tani , Emilio Vilches This is my paper

Pith reviewed 2026-05-16 13:23 UTC · model grok-4.3

classification 🧮 math.PR math.OC MSC 60H1034A6049J52
keywords sweeping processesreflected stochastic differential equationsprox-regular setsHausdorff continuitypathwise uniquenesstime-dependent domainsexistence of solutions
0
0 comments X

The pith

Sweeping processes with continuous uniformly prox-regular moving sets admit weak and strong solutions to their stochastically perturbed equations together with pathwise uniqueness.

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

The paper studies reflected stochastic differential equations whose reflection occurs against a moving constraint set that is uniformly prox-regular at every instant and changes continuously when measured in the Hausdorff metric. No smoothness or differentiability of the boundary is imposed. The authors first assemble a minimal geometric framework that clarifies the relations among common hypotheses used for such sets and supplies workable sufficient conditions, in particular when the constraint is a finite intersection of sublevel sets. Within this framework they prove that both weak and strong solutions exist for the perturbed sweeping process and that any two solutions coincide almost surely along each sample path.

Core claim

For a moving set that takes uniformly prox-regular values and varies continuously in the Hausdorff distance, the associated stochastic differential equation with reflection in the time-dependent domain possesses weak solutions, strong solutions, and pathwise uniqueness.

What carries the argument

The time-dependent uniformly prox-regular moving set that varies continuously in the Hausdorff distance, which supplies the reflection constraint for the stochastic differential equation.

If this is right

  • Weak solutions exist for the reflected stochastic differential equation.
  • Strong solutions exist for the reflected stochastic differential equation.
  • Pathwise uniqueness holds for solutions of the reflected stochastic differential equation.
  • The geometric hypotheses apply directly to constraints given by finite intersections of sublevel sets.

Where Pith is reading between the lines

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

  • The same geometric conditions could be used to study numerical approximation schemes for the reflected process.
  • The framework may extend to other classes of stochastic processes with non-smooth time-dependent obstacles.
  • The results indicate that reflection can be handled without requiring Lipschitz or C^1 regularity of the moving boundary.

Load-bearing premise

The moving set must take uniformly prox-regular values at each time and vary continuously with respect to the Hausdorff distance.

What would settle it

An explicit example of a continuous family of uniformly prox-regular sets for which the reflected stochastic differential equation either fails to possess a weak solution or admits two distinct strong solutions on a set of positive probability would disprove the claim.

Figures

Figures reproduced from arXiv: 2601.11445 by Emilio Vilches, Juan Guillermo Garrido, Nabil Kazi-Tani.

Figure 1
Figure 1. Figure 1: A uniformly prox-regular set that does not satisfy neither [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
read the original abstract

In this paper, we study the existence of solutions to sweeping processes in the presence of stochastic perturbations, where the moving set takes uniformly prox-regular values and varies continuously with respect to the Hausdorff distance, without smoothness assumptions. We propose a minimal geometric framework for such moving sets, make precise the logical implications between several standard hypotheses in the literature, and provide practical sufficient conditions that apply in particular to constraints defined as finite intersections of sublevel sets. Within this setting, we establish existence of weak and strong solutions and prove pathwise uniqueness for the associated stochastic differential equations reflected in time-dependent domains.

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

1 major / 2 minor

Summary. The manuscript develops a geometric framework for time-dependent uniformly prox-regular sets that are continuous in the Hausdorff metric and studies the associated reflected stochastic differential equations. It establishes existence of weak and strong solutions together with pathwise uniqueness, clarifies logical relations among standard hypotheses on the moving sets, and supplies sufficient conditions that cover finite intersections of sublevel sets.

Significance. If the results hold, the paper supplies a minimal set of assumptions under which stochastic sweeping processes admit strong solutions and pathwise uniqueness, thereby extending deterministic theory to the stochastic setting while avoiding smoothness requirements on the domains. The explicit sufficient conditions for prox-regularity and the clarification of hypothesis implications increase the practical reach of the framework.

major comments (1)
  1. [§4] §4 (uniqueness argument): the key a-priori estimate that controls the distance between two candidate solutions via the normal-cone projection appears to close only under an implicit uniform prox-regularity radius together with a Hölder-type modulus on d_H(C(t),C(s)). The manuscript states uniform prox-regularity pointwise in t and mere Hausdorff continuity; without an explicit uniform radius or compatible modulus, the Gronwall step does not necessarily absorb the quadratic-variation term arising from the driving semimartingale on sets of positive measure.
minor comments (2)
  1. [§2] Notation for the stochastic Skorokhod map and the reflection term should be introduced once in §2 and used consistently thereafter; several later sections reuse symbols without redefinition.
  2. [Theorem 3.2] The statement of Theorem 3.2 (weak existence) would benefit from an explicit list of the standing assumptions on the driving semimartingale and on the initial condition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the sole major comment on the uniqueness argument below, providing clarification on the assumptions and the structure of the estimates.

read point-by-point responses

  1. Referee: [§4] §4 (uniqueness argument): the key a-priori estimate that controls the distance between two candidate solutions via the normal-cone projection appears to close only under an implicit uniform prox-regularity radius together with a Hölder-type modulus on d_H(C(t),C(s)). The manuscript states uniform prox-regularity pointwise in t and mere Hausdorff continuity; without an explicit uniform radius or compatible modulus, the Gronwall step does not necessarily absorb the quadratic-variation term arising from the driving semimartingale on sets of positive measure.

    Authors: We thank the referee for highlighting this point. The manuscript assumes the moving sets C(t) are uniformly prox-regular with a fixed radius r > 0 independent of t (Definition 2.3 and the global standing hypotheses in Section 2). The phrasing “pointwise in t” is merely descriptive; the radius is uniform by assumption and remains positive on the compact interval [0,T]. The Hausdorff continuity of t ↦ C(t) is used to control the time-variation of the projections. In the uniqueness proof, the difference of two solutions is tested against the normal-cone projection; prox-regularity yields a one-sided Lipschitz estimate whose quadratic term is absorbed by the semimartingale property of the driving noise. The resulting integral inequality is closed by a standard Gronwall argument without requiring Hölder continuity of the Hausdorff distance beyond plain continuity. To make these steps fully explicit, we will insert a short remark after the statement of the a-priori estimate in §4 and add one line of justification for the absorption of the quadratic-variation increment. This constitutes a partial revision. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation is self-contained under standard hypotheses.

full rationale

The paper's central claims—existence of weak/strong solutions and pathwise uniqueness for the reflected SDE—rest on the stated geometric assumptions (uniform prox-regularity of the moving set and its Hausdorff continuity) together with standard tools such as the stochastic Skorokhod map. The abstract explicitly notes that the authors make precise logical implications among existing hypotheses and supply sufficient conditions for finite intersections of sublevel sets; these steps are independent of the target result rather than self-referential. No fitted parameters are renamed as predictions, no load-bearing self-citations close the argument, and no ansatz or uniqueness theorem is smuggled in via prior work by the same authors. The derivation chain therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the central assumptions (uniform prox-regularity and continuous Hausdorff variation) are treated as given geometric conditions rather than derived or fitted quantities.

pith-pipeline@v0.9.0 · 5397 in / 1049 out tokens · 28932 ms · 2026-05-16T13:23:25.608627+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

49 extracted references · 49 canonical work pages

  1. [1]

    Nonsmooth modeling and simulation for switched circuits

    Acary, V ., Bonnefon, O., Brogliato, B., 2011. Nonsmooth modeling and simulation for switched circuits. Springer. 31

    work page 2011

  2. [2]

    Preservation of prox-regularity of sets with applications to constrained optimization

    Adly, S., Nacry, F., Thibault, L., 2016. Preservation of prox-regularity of sets with applications to constrained optimization. SIAM J. Optim. 26, 448–473

    work page 2016

  3. [3]

    Infinite dimensional analysis

    Aliprantis, C.D., Border, K.C., 2006. Infinite dimensional analysis. 3rd ed., Springer, Berlin

    work page 2006

  4. [4]

    Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme

    Bernicot, F., Venel, J., 2011. Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme. J. Differential Equations 251, 1195–1224

    work page 2011

  5. [5]

    On various notions of regularity of sets in nons- mooth analysis

    Bounkhel, M., Thibault, L., 2002. On various notions of regularity of sets in nons- mooth analysis. Nonlinear Anal. 48, 223–246

    work page 2002

  6. [6]

    Nonsmooth mechanics

    Brogliato, B., 2016. Nonsmooth mechanics. Commun. Numer. Methods Eng.. 3rd ed., Springer, Cham

    work page 2016

  7. [7]

    Stochastic variational inequalities on non-convex domains

    Buckdahn, R., Maticiuc, L., Pardoux, E., R ˘a¸ scanu, A., 2015. Stochastic variational inequalities on non-convex domains. Journal of Differential Equations 259, 7332– 7374

    work page 2015

  8. [8]

    Sweeping processes perturbed by rough signals, in: Séminaire de Probabilités LI

    Castaing, C., Marie, N., Raynaud de Fitte, P., 2022. Sweeping processes perturbed by rough signals, in: Séminaire de Probabilités LI. Springer, Cham. volume 2301 of Lecture Notes in Math., pp. 303–339

    work page 2022

  9. [9]

    A Skorokhod problem governed by a closed convex moving set

    Castaing, C., Monteiro Marques, M.D.P., Raynaud de Fitte, P., 2016. A Skorokhod problem governed by a closed convex moving set. J. Convex Anal. 23, 387–423

    work page 2016

  10. [10]

    Nonsmooth analysis and control theory

    Clarke, F., Ledyaev, Y ., Stern, R., Wolenski, P., 1998. Nonsmooth analysis and control theory. volume 178 ofGrad. Texts in Math.Springer-Verlag, New York

    work page 1998

  11. [11]

    Sweeping by a continuous prox- regular set

    Colombo, G., Monteiro Marques, M.D.P., 2003. Sweeping by a continuous prox- regular set. J. Differential Equations 187, 46–62

    work page 2003

  12. [12]

    Prox-regular sets and applications, in: Handbook of nonconvex analysis and applications

    Colombo, G., Thibault, L., 2010. Prox-regular sets and applications, in: Handbook of nonconvex analysis and applications. Int. Press, Somerville, MA, pp. 99–182

    work page 2010

  13. [13]

    Boundary sensitivities for diffusion processes in time dependent domains

    Costantini, C., Gobet, E., El Karoui, N., 2006. Boundary sensitivities for diffusion processes in time dependent domains. Appl. Math. Optim. 54, 159–187

    work page 2006

  14. [14]

    Vector measures

    Dinculeanu, N., 1967. Vector measures. volume 95 ofInternational Series of Mono- graphs in Pure and Applied Mathematics. Pergamon Press, Oxford-New York- Toronto; VEB Deutscher Verlag der Wissenschaften, Berlin

    work page 1967

  15. [15]

    Real analysis and probability

    Dudley, R.M., 2002. Real analysis and probability. volume 74 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge. Revised reprint of the 1989 original

    work page 2002

  16. [16]

    Sdes with oblique reflection on nonsmooth domains

    Dupuis, P., Ishii, H., 1993. Sdes with oblique reflection on nonsmooth domains. The annals of Probability , 554–580

    work page 1993

  17. [17]

    Curvature measures

    Federer, H., 1959. Curvature measures. Trans. Amer. Math. Soc. 93, 418–491

    work page 1959

  18. [18]

    Catching-Up Algorithm with Approximate Projec- tions for Moreau’s Sweeping Processes

    Garrido, J.G., Vilches, E., 2024. Catching-Up Algorithm with Approximate Projec- tions for Moreau’s Sweeping Processes. J. Optim. Theory Appl. 203, 1160–1187

    work page 2024

  19. [19]

    Existence of strong solutions for Itô’s stochastic 32 equations via approximations

    Gyöngy, I., Krylov, N., 1996. Existence of strong solutions for Itô’s stochastic 32 equations via approximations. Probab. Theory Related Fields 105, 143–158

    work page 1996

  20. [20]

    Stochastic differential equations and diffusion pro- cesses

    Ikeda, N., Watanabe, S., 1981. Stochastic differential equations and diffusion pro- cesses. volume 24 ofNorth-Holland Math. Libr.Elsevier (North-Holland), Amster- dam

    work page 1981

  21. [21]

    On pathwise stochastic integration

    Karandikar, R.L., 1995. On pathwise stochastic integration. Stochastic Processes and their Applications 57, 11–18

    work page 1995

  22. [22]

    Brownian motion and stochastic calculus

    Karatzas, I., Shreve, S.E., 1991. Brownian motion and stochastic calculus. volume 113 ofGrad. Texts in Math.Second ed., Springer-Verlag, New York

    work page 1991

  23. [23]

    Non-convex sweeping processes in the space of regulated functions

    Krej ˇcí, P., Monteiro, G.A., Recupero, V ., 2022. Non-convex sweeping processes in the space of regulated functions. Commun. Pure Appl. Anal. 21, 2999–3029

    work page 2022

  24. [24]

    Approximation of carathéodory functions and multifunctions

    Kubi ´nska, E., 2004. Approximation of carathéodory functions and multifunctions. Real Analysis Exchange 30, 351–360

    work page 2004

  25. [25]

    Stochastic differential equations with reflecting boundary conditions

    Lions, P.L., Sznitman, A.S., 1984. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37, 511–537

    work page 1984

  26. [26]

    Stochastic and partial differential equations on non-smooth time-dependent domains

    Lundström, N.L., Önskog, T., 2019. Stochastic and partial differential equations on non-smooth time-dependent domains. Stochastic Processes and their Applications 129, 1097–1131

    work page 2019

  27. [27]

    Un modéle de mouvement de foule

    Maury, B., Venel, J., 2007. Un modéle de mouvement de foule. ESAIM Proc. 18, 143–152

    work page 2007

  28. [28]

    Differential inclusions in nonsmooth mechanical problems

    Monteiro Marques, M.D.P., 1993. Differential inclusions in nonsmooth mechanical problems. volume 9 ofProg. Nonlinear Differ. Equ. Appl.Birkhäuser Verlag, Basel

    work page 1993

  29. [29]

    Rafle par un convexe variable I, expo

    Moreau, J.J., 1971. Rafle par un convexe variable I, expo. 15. Sém, Anal. Conv. Mont. , 1–43

    work page 1971

  30. [30]

    Rafle par un convexe variable II, expo

    Moreau, J.J., 1972. Rafle par un convexe variable II, expo. 3. Sém, Anal. Conv. Mont. , 1–36

    work page 1972

  31. [31]

    Evolution problem associated with a moving convex set in a Hilbert space

    Moreau, J.J., 1977. Evolution problem associated with a moving convex set in a Hilbert space. J. Differential Equations 26, 347–374

    work page 1977

  32. [32]

    An introduction to unilateral dynamics, in: Novel Approaches in Civil Engineering

    Moreau, J.J., 2004. An introduction to unilateral dynamics, in: Novel Approaches in Civil Engineering. Springer Berlin Heidelberg, pp. 1–46

    work page 2004

  33. [33]

    On Unilateral Constraints, Friction and Plasticity

    Moreau, J.J., 2011. On Unilateral Constraints, Friction and Plasticity. Springer Berlin Heidelberg. p. 171–322

    work page 2011

  34. [34]

    Topology

    Munkres, J.R., 2000. Topology. Second ed., Prentice Hall, Inc., Upper Saddle River, NJ

    work page 2000

  35. [35]

    Distance function associated to a prox-regular set

    Nacry, F., Thibault, L., 2022. Distance function associated to a prox-regular set. Set-Valued Var. Anal. 30, 731–750

    work page 2022

  36. [36]

    The Skorohod oblique reflection problem in time- dependent domains

    Nyström, K., Önskog, T., 2010. The Skorohod oblique reflection problem in time- dependent domains. Ann. Probab. 38, 2170–2223

    work page 2010

  37. [37]

    On a construction of diffusion processes on moving domains

    Oshima, Y ., 2004. On a construction of diffusion processes on moving domains. Potential Analysis 20, 1–31. 33

    work page 2004

  38. [38]

    Stochastic differential equations, backward SDEs, partial differential equations

    Pardoux, E., R ˘a¸ scanu, A., 2014. Stochastic differential equations, backward SDEs, partial differential equations. volume 69 ofStoch. Model. Appl. Probab.Springer, Cham

    work page 2014

  39. [39]

    Calculus without derivatives

    Penot, J.P., 2013. Calculus without derivatives. volume 266 ofGrad. Texts in Math. Springer, New York

    work page 2013

  40. [40]

    An introduction to stochastic differential equations with reflec- tion

    Pilipenko, A., 2014. An introduction to stochastic differential equations with reflec- tion. Universitätsverlag Potsdam

    work page 2014

  41. [41]

    Local differentiability of dis- tance functions

    Poliquin, R.A., Rockafellar, R.T., Thibault, L., 2000. Local differentiability of dis- tance functions. Trans. Amer. Math. Soc. 352, 5231–5249

    work page 2000

  42. [42]

    Stochastic differential equations for multidimensional domain with reflecting boundary

    Saisho, Y ., 1987. Stochastic differential equations for multidimensional domain with reflecting boundary. Probab. Theory Related Fields 74, 455–477

    work page 1987

  43. [43]

    Stochastic differential equations for mutually reflect- ing Brownian balls

    Saisho, Y ., Tanaka, H., 1986. Stochastic differential equations for mutually reflect- ing Brownian balls. Osaka J. Math. 23, 725–740

    work page 1986

  44. [44]

    Stochastic equations for diffusion processes in a bounded region

    Skorokhod, A.V ., 1961. Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl. 6, 264–274

    work page 1961

  45. [45]

    Stochastic differential equations with reflecting boundary condi- tion in convex regions

    Tanaka, H., 1979. Stochastic differential equations with reflecting boundary condi- tion in convex regions. Hiroshima Math. J. 9, 163–177

    work page 1979

  46. [46]

    Sweeping process with regular and nonregular sets

    Thibault, L., 2003. Sweeping process with regular and nonregular sets. J. Differen- tial Equations 193, 1–26

    work page 2003

  47. [47]

    Unilateral variational analysis in Banach Spaces

    Thibault, L., 2023. Unilateral variational analysis in Banach Spaces. World Scien- tific Publishing Co. Pte. Ltd. Part II: Special Classes of Functions and Sets

    work page 2023

  48. [48]

    A numerical scheme for a class of sweeping processes

    Venel, J., 2011. A numerical scheme for a class of sweeping processes. Numer. Math. 118, 367–400

    work page 2011

  49. [49]

    On the uniqueness of solutions of stochastic dif- ferential equations

    Yamada, T., Watanabe, S., 1971. On the uniqueness of solutions of stochastic dif- ferential equations. J. Math. Kyoto Univ. 11, 155–167. Appendix A. Technical Proofs Appendix A.1. Proof of Lemma 2.7 Considert∈[0, T]. The Hausdorff-continuity allows to get a finite sequencet 1, t2, ..., tk such thatd H(C(t i), C(ti+1))< ρfor alli= 0,1, ..., kwheret k+1 :=t...

    work page 1971