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arxiv: 2604.20327 · v1 · submitted 2026-04-22 · 🧮 math.PR

Persistent Homology of the Wiener Sausage II: A Central Limit Theorem for Drifted Planar Brownian Motion

Tristan Guillaume (CYU) This is my paper

Pith reviewed 2026-05-09 22:46 UTC · model grok-4.3

classification 🧮 math.PR
keywords limitcentralsausagetheoremwienerbetabrowniandrift
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The pith

The smoothed Betti-1 curve functional of the radius-r Wiener sausage for drifted planar Brownian motion obeys a central limit theorem with deterministic centering and finite variance.

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

The Wiener sausage is the set of all points within distance r of a random path that Brownian motion traces in the plane, but here the path has a constant drift pushing it in one direction. The authors count the holes inside this growing sausage using the first Betti number, then smooth that count over a range of r values with a test function. Their earlier work showed the smoothed count grows linearly with time on average. This paper shows the deviations from that average, divided by the square root of time, settle into a bell-shaped normal curve. The proof re-uses a regeneration trick that cuts the path into independent cycles along the drift direction. It adds a new L2 analysis of the topological changes that happen exactly at those cut points and proves a polynomial moment bound on the hole counts so that the variance stays finite. The result also extends to several test functions at once, giving a joint Gaussian limit.

Core claim

There exist a deterministic constant ρ_ψ and a variance σ_ψ² ≥ 0 such that (Φ_ψ(t) − ρ_ψ t)/√t converges in distribution to N(0, σ_ψ²) as t → ∞, where Φ_ψ(t) is the smoothed integral of the first Betti number of the Wiener sausage.

Load-bearing premise

A finite-time polynomial moment bound holds for the integrated hole counts of the Wiener sausage; this bound is invoked to obtain square-integrability of cycle increments, within-cycle oscillations, and the last incomplete cycle, enabling the CLT for 1-dependent sequences after the renewal time change.

read the original abstract

Let $X_t = B_t + \mu t$, $t \geq 0$, be planar Brownian motion with nonzero drift, and let $K_t^r = \{x \in \mathbb{R}^2 : {\rm dist}(x, X[0,t]) \leq r\}$ be the radius-$r$ Wiener sausage up to time $t$. For a bounded Borel function $\psi$ supported in a compact interval $[r_0, r_1] \subset (0,\infty)$, consider the smoothed Betti-curve functional $\Phi_\psi(t) := \int_{r_0}^{r_1} \beta_1^t(r)\,\psi(r)\,dr$, where $\beta_1^t(r)$ denotes the number of holes of $K_t^r$. In a previous paper, a regeneration scheme along the drift direction was used to prove a law of large numbers for $\Phi_\psi(t)$. In the present paper we prove the corresponding central limit theorem. More precisely, there exist a deterministic constant $\rho_\psi$ and a variance $\sigma_\psi^2 \geq 0$ such that $(\Phi_\psi(t) - \rho_\psi t)/\sqrt{t} \xrightarrow{d}_{t \to \infty} \mathcal{N}(0, \sigma_\psi^2)$. We also obtain the finite-dimensional Gaussian limit for finitely many test functions. The proof preserves the regenerative structure of the law of large numbers, but requires a new $L^2$ analysis of the topological interface terms created at regeneration cuts. The key input is a finite-time polynomial moment bound for integrated hole counts of the Wiener sausage. This yields square-integrability of cycle increments, within-cycle oscillations, and the last incomplete-cycle remainder, which in turn allows one to combine a standard central limit theorem for stationary $1$-dependent sequences with a renewal time-change argument.

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.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claim rests on standard properties of drifted Brownian motion, the regeneration scheme introduced in the authors' earlier paper, and a new finite-time polynomial moment bound on integrated hole counts that is treated as a key technical input whose proof is not detailed in the abstract.

axioms (3)
  • standard math Standard properties of planar Brownian motion with nonzero drift
    Used to define the process X_t and the sausage K_t^r throughout the argument.
  • domain assumption Existence of a regeneration scheme along the drift direction
    Invoked from the previous paper to decompose the path into cycles for both the LLN and the CLT.
  • ad hoc to paper Finite-time polynomial moment bound for integrated hole counts
    Stated as the key input that yields square-integrability of increments and remainders; its validity is required for the L2 analysis.

pith-pipeline@v0.9.0 · 5654 in / 1709 out tokens · 27174 ms · 2026-05-09T22:46:38.978008+00:00 · methodology

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Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

  1. [1]

    Asmussen, Applied Probability and Queues , 2nd ed., Springer , New York, 2003

    S. Asmussen, Applied Probability and Queues , 2nd ed., Springer , New York, 2003

    work page 2003

  2. [2]

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

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

    work page 1999

  3. [3]

    Bobrowski and M

    O. Bobrowski and M. Kahle, Topology of random geometric c omplexes: a survey , J. Appl. Comput. Topol. 1 (2018), 331–364

    work page 2018

  4. [4]

    R. C. Bradley , Basic properties of strong mixing conditi ons. A survey and some open ques- tions, Probab. Surv . 2 (2005), 107–144

    work page 2005

  5. [5]

    Chazal, V

    F . Chazal, V . de Silva, M. Glisse, and S. Oudot, The Structure and Stability of Persistence Modules, Springer , 2016

    work page 2016

  6. [6]

    Cohen-Steiner , H

    D. Cohen-Steiner , H. Edelsbrunner , and J. Harer , Stability of persistence diagrams, Discrete Comput. Geom. 37 (2007), 103–120

    work page 2007

  7. [7]

    Feller ,An Introduction to Probability Theory and Its Applications , Vol

    W . Feller ,An Introduction to Probability Theory and Its Applications , Vol. II , 2nd ed., Wiley , New York, 1971

    work page 1971

  8. [8]

    Persistence of the Wiener Sausage: Sampling Stability and a Law of Large Numbers for Drifted Planar Brownian Motion DRAFT -CURRENTLY UNDER REVIEW

    T . Guillaume, Persistence of the Wiener Sausage: Sampling Stability and a Law of Large Numbers for Drifted Planar Brownian Motion , arXiv:2604.03130

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    Hiraoka, T

    Y . Hiraoka, T . Shirai, and K. D. Trinh, Limit theorems for persistence diagrams, Ann. Appl. Probab. 28 (2018), 2740–2780

    work page 2018

  10. [10]

    Hoeffding and H

    W . Hoeffding and H. Robbins, The central limit theorem f or dependent random variables, Duke Math. J. 15 (1948), 773–780

    work page 1948

  11. [11]

    Krebs and C

    J. Krebs and C. Hirsch, Functional central limit theore ms for persistent Betti numbers on cylindrical networks, Scand. J. Stat. 49 (2022), 427–454

    work page 2022

  12. [12]

    J. T . N. Krebs and W . Polonik, On the asymptotic normalit y of persistent Betti numbers, Electron. J. Stat. 16 (2022), 2551–2588

    work page 2022

  13. [13]

    Sznitman, Brownian Motion, Obstacles and Random Media , Springer , Berlin, 1998

    A.-S. Sznitman, Brownian Motion, Obstacles and Random Media , Springer , Berlin, 1998

    work page 1998

  14. [14]

    Whitt, Stochastic-Process Limits , Springer , New York, 2002

    W . Whitt, Stochastic-Process Limits , Springer , New York, 2002. Page 12/12

    work page 2002