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arxiv: 2510.24593 · v1 · submitted 2025-10-28 · 🧮 math.PR · math.DG

Brownian motion on spaces of discrete regular curves

Karen Habermann , Emmanuel Hartman This is my paper

Pith reviewed 2026-05-18 03:01 UTC · model grok-4.3

classification 🧮 math.PR math.DG
keywords stochastic completenessBrownian motiondiscrete regular curvesSobolev metricsgeodesic completenessshape analysisvolume growthGrigor'yan theorem
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The pith

Geodesically complete spaces of discrete regular curves with order-two or higher Sobolev metrics are also stochastically complete.

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

The paper shows that spaces of discrete regular curves in Euclidean space become stochastically complete precisely when they are geodesically complete under discrete Sobolev-type metrics of order two or higher. This follows from bounding the volume growth of geodesic balls to meet the hypotheses of Grigor'yan's theorem. A sympathetic reader would care because the result supplies a foundation for running Brownian motion indefinitely on the full space of curves, which supports concrete statistical tasks such as data inference and imputation. The work also supplies simulations of sample paths and heuristics suggesting possible completeness for lower-order metrics in special cases like triangles.

Core claim

By relying on a general result by Grigor'yan and controlling the volume growth of geodesic balls, the authors establish that all spaces of discrete regular curves that are geodesically complete are also stochastically complete, meaning the associated Brownian motion exists for all times.

What carries the argument

Grigor'yan's theorem on stochastic completeness, applied after establishing volume growth bounds for geodesic balls in the spaces of discrete regular curves.

If this is right

  • This provides a rigorous footing for performing data statistics such as data inference and data imputation on these spaces.
  • It is the first stochastic completeness result in shape analysis that applies to the full shape space of interest.
  • Simulations for sample paths of Brownian motion are included for illustration.
  • Heuristics suggest that the space of triangles remains stochastically complete even for Sobolev-type metrics of order zero and one.

Where Pith is reading between the lines

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

  • The volume growth control technique could be tested on discrete approximations to other infinite-dimensional shape spaces.
  • Eternal Brownian motion on these spaces could support generative sampling or diffusion models for random curves in applications.
  • A full proof extending the heuristics to order-one metrics would enlarge the set of usable metrics without raising the order.

Load-bearing premise

The volume growth of geodesic balls in these spaces can be controlled sufficiently to satisfy the hypotheses of Grigor'yan's theorem.

What would settle it

An explicit volume estimate for geodesic balls that grows fast enough to violate the integral condition in Grigor'yan's theorem, or a simulation in which Brownian paths explode in finite time on one of the geodesically complete spaces.

Figures

Figures reproduced from arXiv: 2510.24593 by Emmanuel Hartman, Karen Habermann.

Figure 1
Figure 1. Figure 1: Three examples of Brownian motion on the space of discrete regular curves with respect [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Brownian motion on the space of discrete regular curves in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plots of the conformal factors f0, f1 and f2 corresponding to the restrictions of the metrics g 0 , g 1 and g 2 to the space of triangles with two fixed vertices. We display the conformal factors on [−2, 2] × [−2, 2] as well as a small region around the singularity at v0 = (1, 0). As each iteration in the definition of the metric g m introduces one additional division by |e0(v)|, we expect that, more gener… view at source ↗
read the original abstract

We introduce and study Brownian motion on spaces of discrete regular curves in Euclidean space equipped with discrete Sobolev-type metrics. It has been established that these spaces of discrete regular curves are geodesically complete if and only if the Sobolev-type metric is of order two or higher. By relying on a general result by Grigor'yan and controlling the volume growth of geodesic balls, we show that all spaces of discrete regular curves that are geodesically complete are also stochastically complete, that is, the associated Brownian motion exists for all times. This provides a rigorous footing for performing data statistics, such as data inference and data imputation, on these spaces. Our result is the first stochastic completeness result in shape analysis that applies to the full shape space of interest. For illustrative purposes, we include simulations for sample paths of Brownian motion on spaces of discrete regular curves. For the space of triangles in the plane modulo rotation, translation and scaling, we further provide heuristics which suggest that this space remains stochastically complete even for Sobolev-type metrics of order zero and one.

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 introduces spaces of discrete regular curves in Euclidean space equipped with discrete Sobolev-type metrics and studies the associated Brownian motion. It establishes that these spaces are geodesically complete if and only if the metric order is at least two. By controlling the volume growth of geodesic balls and invoking Grigor'yan's theorem on stochastic completeness for complete length spaces, the authors conclude that geodesic completeness implies stochastic completeness (Brownian motion exists for all times). The result is positioned as the first stochastic completeness theorem in shape analysis applying to the full shape space of interest; the paper includes illustrative simulations of sample paths and heuristics suggesting stochastic completeness may hold for order-zero and order-one metrics on the space of triangles modulo similarity.

Significance. If the volume-growth control is carried out with explicit, quantitative bounds derived from the discrete Sobolev metric that verify Grigor'yan's integral-divergence condition, the result would supply the first rigorous justification for long-time Brownian motion on these infinite-dimensional shape spaces, thereby grounding statistical procedures such as data inference and imputation.

major comments (1)
  1. [Section applying Grigor'yan's theorem (volume-growth control)] The central argument applies Grigor'yan's theorem after asserting control of the volume growth of geodesic balls, yet the manuscript supplies no explicit upper bound on Vol(B(r)) (e.g., polynomial, exponential, or exp(o(r^2))) expressed in terms of the discrete Sobolev metric of order ≥2, nor any verification that ∫_1^∞ r / log Vol(B(r)) dr diverges. Because geodesic completeness alone does not guarantee the required growth restriction, this step is load-bearing for the stochastic-completeness claim.
minor comments (2)
  1. [Final paragraph on triangles] The heuristics for stochastic completeness at orders 0 and 1 on the triangle space are clearly labeled as non-rigorous, but a short remark distinguishing them from the main theorem would improve readability.
  2. [Simulations paragraph] Simulation details (time-stepping scheme, truncation of the infinite-dimensional space, and convergence diagnostics) are only sketched; adding a brief description or reference to the numerical method would aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point regarding the rigor of the volume-growth argument. We address the comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [Section applying Grigor'yan's theorem (volume-growth control)] The central argument applies Grigor'yan's theorem after asserting control of the volume growth of geodesic balls, yet the manuscript supplies no explicit upper bound on Vol(B(r)) (e.g., polynomial, exponential, or exp(o(r^2))) expressed in terms of the discrete Sobolev metric of order ≥2, nor any verification that ∫_1^∞ r / log Vol(B(r)) dr diverges. Because geodesic completeness alone does not guarantee the required growth restriction, this step is load-bearing for the stochastic-completeness claim.

    Authors: We agree with the referee that the manuscript asserts control of the volume growth of geodesic balls but does not supply an explicit upper bound on Vol(B(r)) expressed in terms of the discrete Sobolev metric (for order ≥2) nor an explicit verification that the integral ∫_1^∞ r / log Vol(B(r)) dr diverges. While geodesic completeness is established separately, the referee is correct that this does not automatically imply the growth condition needed for Grigor'yan's theorem. In the revised manuscript we will derive and insert the required quantitative upper bound on Vol(B(r)) directly from the discrete Sobolev metric of order ≥2 and verify that the integral diverges, thereby completing the application of the theorem. revision: yes

Circularity Check

0 steps flagged

No circularity: external Grigor'yan theorem applied after independent volume-growth control from discrete metric

full rationale

The derivation invokes Grigor'yan's external theorem on stochastic completeness for geodesically complete length spaces whose geodesic balls satisfy a divergent integral condition on volume growth. The paper states that this volume growth is controlled by the discrete Sobolev-type metric of order two or higher (the same condition already used for geodesic completeness). This control is derived from the metric definition on the space of discrete curves and does not reduce to a self-definition, fitted parameter, or self-citation chain. Grigor'yan's result is a general theorem from geometric analysis, not authored by the present writers, and the application is therefore independent support rather than circular. No load-bearing step collapses to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof depends on an external theorem and an unstated but load-bearing control of volume growth; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Grigor'yan's general result linking geodesic completeness and volume growth to stochastic completeness
    Directly invoked to conclude existence of Brownian motion for all times once volume growth is controlled.

pith-pipeline@v0.9.0 · 5705 in / 1165 out tokens · 42494 ms · 2026-05-18T03:01:02.247277+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    Bauer, C

    M. Bauer, C. Maor, and P. W. Michor. Sobolev metrics on spaces of manifold-valued curves. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, 24(4):1895–1948, 2023

    work page 1948

  2. [2]

    Beutler, F

    S. Beutler, F. Hartwig, M. Rumpf, and B. Wirth. Discrete geodesic calculus in the space of Sobolev curves. arXiv:2505.10298, 15 May 2025. 10

    work page arXiv 2025

  3. [3]

    Bruveris

    M. Bruveris. Completeness properties of Sobolev metrics on the space of curves.Journal of Geometric Mechanics, 7(2):125–150, 2015

    work page 2015

  4. [4]

    Bruveris, P

    M. Bruveris, P. W. Michor, and D. Mumford. Geodesic completeness for Sobolev metrics on the space of immersed plane curves. InForum of Mathematics, Sigma, volume 2. Cambridge University Press, 2014

    work page 2014

  5. [5]

    Cerqueira, E

    J. Cerqueira, E. Hartman, E. Klassen, and M. Bauer. Sobolev metrics on spaces of discrete regular curves.Discrete and Continuous Dynamical Systems. Series A, 45(12):5154–5177, 2025

    work page 2025

  6. [6]

    Grigor’yan

    A. Grigor’yan. Stochastically complete manifolds.Doklady Akademii Nauk SSSR, 290(3):534– 537, 1986

    work page 1986

  7. [7]

    Grigor’yan

    A. Grigor’yan. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds.Bulletin (New Series) of the American Mathe- matical Society, 36(2):135–249, 1999

    work page 1999

  8. [8]

    Habermann, P

    K. Habermann, P. Harms, and S. Sommer. Long-time existence of Brownian motion on con- figurations of two landmarks.Bulletin of the London Mathematical Society, 56(5):1658–1679, 2024

    work page 2024

  9. [9]

    Habermann, S

    K. Habermann, S. C. Preston, and S. Sommer. Characterization of geodesic completeness for landmark space. arXiv:2503.10611, 13 March 2025

    work page arXiv 2025

  10. [10]

    P. W. Michor and D. Mumford. Riemannian geometries on spaces of plane curves.Journal of the European Mathematical Society, 8(1):1–48, 2006

    work page 2006

  11. [11]

    P. W. Michor and D. Mumford. An overview of the Riemannian metrics on spaces of curves us- ing the Hamiltonian approach.Applied and Computational Harmonic Analysis, 23(1):74–113, 2007

    work page 2007

  12. [12]

    Nenciu and I

    G. Nenciu and I. Nenciu. Drift-diffusion equations on domains inR d: essential self-adjointness and stochastic completeness.Journal of Functional Analysis, 273(8):2619–2654, 2017. 11

    work page 2017