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arxiv: 2604.06708 · v1 · submitted 2026-04-08 · 🧮 math.OC · cs.SY· eess.SY

Uncertainty Propagation in Stochastic Hybrid Systems with Dimension-Varying Resets

Tejaswi K. C. , Taeyoung Lee This is my paper

Pith reviewed 2026-05-10 18:08 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords stochastic hybrid systemsprobability density evolutiondimension-varying resetsweak-form formulationpushforward of fluxuncertainty propagationreset mapsguard sets
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The pith

A weak-formulation using pushforward of probability flux models uncertainty propagation in stochastic hybrid systems where resets change state dimension.

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

The paper establishes a unified weak-form description for how probability densities evolve in stochastic hybrid systems when reset maps alter the dimension of the continuous state. Standard approaches using boundary conditions fall short when resets send probability mass into interiors or onto lower-dimensional sets. By representing reset transfer as the pushforward of probability flux across the guard, the framework handles both dimension-decreasing cases, which produce interior source densities, and dimension-increasing cases, which produce singular sources. This matters for accurate modeling of systems involving merging or splitting states, such as particles combining and separating.

Core claim

The central claim is that reset-induced probability transfer in dimension-varying stochastic hybrid systems can be captured by the pushforward of probability flux across the guard set in a weak formulation. This yields a unified description where dimension reduction appears as an interior source density and dimension increase as a singular source on a lower-dimensional subset.

What carries the argument

The pushforward of probability flux across the guard, which represents reset-induced transfer in the weak form.

If this is right

  • When the reset decreases dimension, transferred probability appears as an interior source density.
  • When the reset increases dimension, it appears as a singular source supported on a lower-dimensional subset.
  • The framework provides a unified description beyond classical boundary-condition-based formulations for such systems.
  • The approach is demonstrated on a stochastic hybrid model of two particles merging into one and later splitting back into two.

Where Pith is reading between the lines

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

  • Numerical schemes solving the resulting weak equations could be developed to propagate uncertainty in merging or splitting dynamics.
  • The same flux-pushforward idea might apply to other hybrid systems where probability mass concentrates on lower-dimensional sets.
  • Verification on the merge-split example would show where boundary-condition methods miss the generated source terms.

Load-bearing premise

The reset maps must be sufficiently regular for the pushforward of the probability flux to be well-defined, and the system must admit a probability density without unmodeled singularities.

What would settle it

Compute the probability density evolution for the two-particle merge-and-split model using both the proposed weak formulation and standard boundary conditions, then check whether the densities differ in the presence of dimension-changing resets.

Figures

Figures reproduced from arXiv: 2604.06708 by Taeyoung Lee, Tejaswi K. C..

Figure 1
Figure 1. Figure 1: Illustration of the continuous state space [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Vector field on X1 follows the Fokker–Planck equation on X2 with reflecting boundary at xC = 0, vanishing trace at the guard xC = 1, and the source term s2 induced by outgoing probability flux from Mode 1. This example highlights two fundamentally different types of reset-induced source terms. When probability flows from Mode 1 to Mode 2, the pushforward of boundary flux produces an absolutely continuous s… view at source ↗
Figure 4
Figure 4. Figure 4: Monte Carlo density evolution for Mode 1 (left) and [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Probability mass evolution (solid: proposed hybrid [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
read the original abstract

This paper studies probability density evolution for stochastic hybrid systems with reset maps that change the dimension of the continuous state across modes. Existing Frobenius--Perron formulations typically represent reset-induced probability transfer through boundary conditions, which is insufficient when resets map guard sets into the interior or onto lower-dimensional subsets of another mode. We develop a weak-form formulation in which reset-induced transfer is represented by the pushforward of probability flux across the guard, yielding a unified description for such systems. The proposed framework naturally captures both cases: when the reset decreases dimension, the transferred probability appears as an interior source density, whereas when the reset increases dimension, it generally appears as a singular source supported on a lower-dimensional subset. The approach is illustrated using a stochastic hybrid model in which two particles merge into one and later split back into two, demonstrating how dimension-changing resets lead to source terms beyond classical boundary-condition-based formulations.

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

2 major / 2 minor

Summary. The paper develops a weak-form formulation for probability density evolution in stochastic hybrid systems with reset maps that change the dimension of the continuous state. Reset-induced transfer is represented via pushforward of probability flux across guards, providing a unified description that captures dimension-decreasing resets as interior source densities and dimension-increasing resets as singular sources on lower-dimensional subsets. This is positioned as an improvement over traditional Frobenius-Perron or boundary-condition approaches, and demonstrated on a stochastic hybrid model of two particles merging into one and later splitting back into two.

Significance. If rigorously supported, the framework would extend uncertainty propagation tools to a broader class of hybrid systems involving coalescence or fission, where state dimension varies. This could enable more accurate modeling in applications such as multi-agent coordination or biological processes, by handling probability transfers that existing fixed-dimension methods cannot represent without ad-hoc adjustments. The particle example highlights the practical distinction from classical formulations.

major comments (2)
  1. [Abstract and weak-formulation section] Abstract and weak-formulation section: The central claim that the pushforward yields a 'unified description' for dimension-increasing resets (appearing as singular sources) is load-bearing, yet the formulation is framed around evolution of a probability density. When the image of the guard is supported on a lower-dimensional subset, the overall measure is no longer absolutely continuous w.r.t. Lebesgue measure on the target space; the weak form tested against smooth compactly supported functions on the full-dimensional domain therefore requires an explicit decomposition into absolutely continuous plus singular parts, or the source term remains incompletely specified.
  2. [Particle merge-split example] Particle merge-split example: The illustration demonstrates dimension-changing resets but supplies no explicit equations for the resulting source terms, no error bounds, and no verification (e.g., against particle simulations or Monte Carlo) that the weak formulation correctly reproduces the probability transfer after the first dimension-increasing reset. This leaves the practical advantage over boundary-condition methods unquantified.
minor comments (2)
  1. The abstract would benefit from a concise statement of the regularity assumptions on the reset maps and guard sets that ensure the pushforward of probability flux is well-defined.
  2. Notation for the probability flux and its pushforward should be introduced with a clear reference to the underlying measure-theoretic construction to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript accordingly to strengthen the presentation of the weak formulation and the illustrative example.

read point-by-point responses

  1. Referee: [Abstract and weak-formulation section] Abstract and weak-formulation section: The central claim that the pushforward yields a 'unified description' for dimension-increasing resets (appearing as singular sources) is load-bearing, yet the formulation is framed around evolution of a probability density. When the image of the guard is supported on a lower-dimensional subset, the overall measure is no longer absolutely continuous w.r.t. Lebesgue measure on the target space; the weak form tested against smooth compactly supported functions on the full-dimensional domain therefore requires an explicit decomposition into absolutely continuous plus singular parts, or the source term remains incompletely specified.

    Authors: We appreciate the referee's observation on the measure-theoretic requirements. The weak formulation is developed for general probability measures (not restricted to densities), with the evolution equation understood in the distributional sense. The pushforward of the probability flux across the guard defines the source as a Radon measure that may be singular. In the revised manuscript, we will explicitly introduce the decomposition of the probability measure into absolutely continuous and singular parts and clarify how the weak form, tested against C^∞_c test functions, incorporates the singular source via the pushforward. This makes the unified description complete and rigorous. revision: yes

  2. Referee: [Particle merge-split example] Particle merge-split example: The illustration demonstrates dimension-changing resets but supplies no explicit equations for the resulting source terms, no error bounds, and no verification (e.g., against particle simulations or Monte Carlo) that the weak formulation correctly reproduces the probability transfer after the first dimension-increasing reset. This leaves the practical advantage over boundary-condition methods unquantified.

    Authors: We agree that explicit equations for the source terms would enhance the example. In the revision, we will derive and include the explicit expression for the singular source term generated by the dimension-increasing split reset, obtained directly from the pushforward of the incoming probability flux. The example is primarily illustrative of the conceptual distinction from boundary-condition approaches. We will add a brief discussion of how numerical verification could be performed via Monte Carlo simulation of the hybrid process, but we do not include quantitative error bounds or full verification in this theoretical paper, as that would constitute a separate computational study. The practical advantage is demonstrated by the framework's ability to represent singular sources without ad-hoc adjustments. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from measure-theoretic pushforward

full rationale

The paper derives a weak-form evolution equation for probability densities in stochastic hybrid systems by representing reset transfers via the pushforward of probability flux across guards. This construction follows directly from standard measure theory and the definition of the pushforward operator on measures, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The abstract and description explicitly distinguish interior sources (dimension decrease) from singular sources (dimension increase) as consequences of the pushforward, not as inputs that define the framework. The example of merging/splitting particles serves as illustration rather than a tautological fit. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, new entities, or ad-hoc axioms are stated. The framework implicitly relies on standard assumptions of stochastic processes and measure theory.

axioms (2)
  • domain assumption Existence of a probability density for the continuous state in each mode of the hybrid system.
    Required to study probability density evolution via the proposed weak form.
  • domain assumption Reset maps admit a well-defined pushforward operation on probability measures or fluxes.
    Central to representing reset-induced transfer without boundary conditions.

pith-pipeline@v0.9.0 · 5457 in / 1435 out tokens · 137938 ms · 2026-05-10T18:08:48.171946+00:00 · methodology

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Works this paper leans on

11 extracted references · 11 canonical work pages

  1. [1]

    Toward a general theory of stochastic hybrid systems,

    M. L. Bujorianu and J. Lygeros, “Toward a general theory of stochastic hybrid systems,” inStochastic hybrid systems. Springer, 2006, pp. 3–30

    work page 2006

  2. [2]

    Rare event estimation for a large-scale stochastic hybrid system with air traffic application,

    H. A. Blom, G. Bakker, and J. Krystul, “Rare event estimation for a large-scale stochastic hybrid system with air traffic application,”Rare event simulation using Monte Carlo methods, pp. 193–214, 2009

    work page 2009

  3. [3]

    Fluid limit theorems for stochastic hybrid systems with application to neuron models,

    K. Pakdaman, M. Thieullen, and G. Wainrib, “Fluid limit theorems for stochastic hybrid systems with application to neuron models,” Advances in Applied Probability, vol. 42, no. 3, pp. 761–794, 2010

    work page 2010

  4. [4]

    Stochastic hybrid systems: Application to commu- nication networks,

    J. P. Hespanha, “Stochastic hybrid systems: Application to commu- nication networks,” inInternational Workshop on Hybrid Systems: Computation and Control. Springer, 2004, pp. 387–401

    work page 2004

  5. [5]

    A study of the long-term behavior of hybrid systems with symmetries via reduction and the Frobenius–Perron operator,

    M. Oprea, A. Shaw, R. Huq, K. Iwasaki, D. Kassabova, and W. Clark, “A study of the long-term behavior of hybrid systems with symmetries via reduction and the Frobenius–Perron operator,”SIAM Journal on Applied Dynamical Systems, vol. 23, no. 4, pp. 2899–2938, 2024

    work page 2024

  6. [6]

    Spectral Bayesian estimation for general stochastic hybrid systems,

    W. Wang and T. Lee, “Spectral Bayesian estimation for general stochastic hybrid systems,”Automatica, vol. 117, Jul. 2020

    work page 2020

  7. [7]

    Uncertainty propagation for general stochastic hybrid systems on compact Lie groups,

    ——, “Uncertainty propagation for general stochastic hybrid systems on compact Lie groups,”SIAM Journal on Applied Dynamical Systems, vol. 21, no. 3, pp. 2215–2240, Jun. 2022

    work page 2022

  8. [8]

    Uncertainty propagation of stochastic hybrid systems: a case study for types of jump,

    T. C, W. Clark, and T. Lee, “Uncertainty propagation of stochastic hybrid systems: a case study for types of jump,” inProceedings of the IFAC Symposium on Nonlinear Control Systems, Jul. 2025

    work page 2025

  9. [9]

    Øksendal,Stochastic differential equations

    B. Øksendal,Stochastic differential equations. Springer, 2003

    work page 2003

  10. [10]

    V . I. Bogachev and M. A. S. Ruas,Measure theory. Springer, 2007, vol. 1, no. 1

    work page 2007

  11. [11]

    R. J. LeVeque,Finite volume methods for hyperbolic problems. Cambridge university press, 2002, vol. 31

    work page 2002