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arxiv: 2604.03130 · v1 · submitted 2026-04-03 · 🧮 math.PR

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

Tristan Guillaume This is my paper

Pith reviewed 2026-05-13 18:32 UTC · model grok-4.3

classification 🧮 math.PR MSC 60J6555N31
keywords persistent homologyWiener sausageBrownian motion with driftlaw of large numbersregenerationBetti numbersoffset filtrationtopological complexity
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The pith

The smoothed persistence functional of the Wiener sausage for drifted planar Brownian motion converges almost surely and in L1 to a deterministic constant times time.

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

The paper establishes a law of large numbers for the number of holes appearing in the radius-r thickened trace of planar Brownian motion with nonzero drift. Projecting the path onto the drift axis produces a one-dimensional process with positive drift whose ladder times generate independent path segments. A boundary control lemma bounds the change in hole count across segment junctions by combining a Mayer-Vietoris estimate with an area integral from the coarea formula. Once these controls are in place, the time-averaged smoothed count of holes stabilizes at a deterministic positive rate for any weight supported away from radius zero. This linear growth supplies an asymptotic intensity measure for the topological complexity of the sausage.

Core claim

For every bounded Borel weight ψ supported on a compact radius window [r0, r1] with r0 > 0, the smoothed persistence functional Φ_ψ(T), defined by integrating the first Betti number β_T^1(r) against ψ, satisfies Φ_ψ(T)/T → ρ_ψ almost surely and in L1, where ρ_ψ is a deterministic constant. The proof rests on a regeneration decomposition of the path into i.i.d. blocks together with a Boundary Lemma that controls non-additivity of topology under concatenation.

What carries the argument

A regeneration scheme along the drift direction that decomposes the path into i.i.d. blocks via ladder hits of the projected drifted Brownian motion, with topological non-additivity controlled by the Boundary Lemma (Mayer-Vietoris estimate plus coarea formula relating integrated Betti numbers to sausage area).

Load-bearing premise

The regeneration along the drift direction produces i.i.d. path blocks whose changes in hole count at junctions remain uniformly controllable by deterministic geometric bounds from the coarea formula.

What would settle it

Numerical computation of Φ_ψ(T)/T for successively larger T that fails to stabilize at a single deterministic value independent of the realized path.

read the original abstract

We study the persistent homology of the offset filtration generated by the range of a planar Brownian motion with constant nonzero drift. The members of this filtration are the Wiener sausages of increasing radius, and the degree-one persistence diagram records the birth and death of holes in the thickened trace as the radius varies. Our first result is a sampling theorem: for any continuous path in R d observed on a time grid $\pi$n the bottleneck distance between the persistence diagram of the continuous offset filtration and that of the sampled point cloud is bounded by the pathwise modulus of continuity $\omega$X (|$\pi$n|). For Brownian motion this yields the almost-sure rate O |$\pi$n| log(1/|$\pi$n|) . Our second and main result is a law of large numbers for the drifted planar case. For every bounded Borel weight $\psi$ supported on a compact radius window [r0, r1] with r0 > 0, the smoothed persistence functional $\Phi$ $\psi$ (T ), where $\beta$ T 1 (r) counts the holes in the radius-r sausage at time T , satisfies $\Phi$ $\psi$ (T )/T $\rightarrow$ $\rho$ $\psi$ almost surely and in L 1 for a deterministic constant $\rho$ $\psi$ . This yields a finite positive intensity measure on the radius axis that governs the linear growth of topological complexity. The proof introduces a regeneration scheme along the drift direction: projecting the planar path onto the drift axis produces a one-dimensional Brownian motion with positive drift, whose ladder hits and bounded-backtracking events generate i.i.d. path blocks. The non-additivity of topology under concatenation is controlled by a Boundary Lemma, which combines a deterministic Mayer-Vietoris estimate with a geometric bound relating integrated Betti numbers to sausage area via the coarea formula. A Betti-curve representation converts the two-parameter persistence problem into a one-parameter family of fixed-radius hole counts, making the regeneration argument possible.

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 proves a sampling stability theorem for persistence diagrams of offset filtrations on continuous paths in R^d, with the bottleneck distance controlled by the path modulus of continuity (yielding an a.s. rate O(|π_n| log(1/|π_n|)) for Brownian motion), and establishes a law of large numbers for the smoothed persistence functional Φ_ψ(T) associated to the degree-1 persistence diagram of the Wiener sausage of a planar Brownian motion with nonzero drift. For any bounded Borel weight ψ supported on a compact interval [r0,r1] with r0>0, Φ_ψ(T)/T converges almost surely and in L1 to a deterministic constant ρ_ψ. The proof relies on a regeneration scheme that decomposes the path into i.i.d. blocks via ladder times of the projected one-dimensional drifted Brownian motion, together with a Boundary Lemma that controls the non-additivity of Betti numbers at block junctions via a Mayer-Vietoris estimate and the coarea formula.

Significance. If the central claims hold, the work supplies the first rigorous LLN for topological complexity measures of drifted Wiener sausages, converting a two-parameter persistence problem into a one-parameter renewal-reward setting. The regeneration-plus-Boundary-Lemma strategy is a concrete technical contribution that may extend to other path functionals in stochastic geometry.

minor comments (3)
  1. [Abstract / §3] The abstract states that the Boundary Lemma combines a deterministic Mayer-Vietoris estimate with a geometric bound via the coarea formula, but the precise statement of the lemma (including the exact form of the discrepancy term) should appear explicitly in the main text before the renewal argument is invoked.
  2. [Introduction] Notation for the smoothed functional Φ_ψ(T) and the Betti curve β_T^1(r) is introduced only in the abstract; a short paragraph in the introduction defining these objects and the radius window [r0,r1] would improve readability.
  3. [§2] The sampling theorem is stated for general continuous paths in R^d; it would be useful to record whether the same modulus-of-continuity bound holds verbatim for the drifted planar case used in the LLN, or whether an extra logarithmic factor appears.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and accurate summary of our results on the sampling stability theorem for persistence diagrams of offset filtrations and the law of large numbers for the smoothed persistence functional of the drifted planar Wiener sausage. We appreciate the recognition that the regeneration scheme via ladder times, combined with the Boundary Lemma using Mayer-Vietoris and the coarea formula, converts the two-parameter problem into a renewal-reward setting. The recommendation for minor revision is noted; since no specific major comments were raised in the report, we have no substantive points to rebut or revise at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The LLN is obtained from i.i.d. regeneration blocks via the strong Markov property at ladder times, combined with deterministic geometric control from the Boundary Lemma (Mayer-Vietoris plus coarea formula). These yield finite per-block expectations under the regeneration measure, after which standard renewal-reward theory produces the a.s. and L1 convergence to the deterministic intensity ρ_ψ. No fitted parameters, self-definitional reductions, or load-bearing self-citations appear in the argument; the derivation is self-contained against external probabilistic and geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard properties of Brownian motion with drift and classical topological inequalities; no free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Planar Brownian motion with constant nonzero drift has positive speed along the drift axis and admits ladder-height regeneration
    Invoked to generate i.i.d. path blocks
  • standard math Mayer-Vietoris inequality bounds the change in Betti numbers under union of sets
    Used inside the Boundary Lemma

pith-pipeline@v0.9.0 · 5681 in / 1287 out tokens · 40021 ms · 2026-05-13T18:32:43.091350+00:00 · methodology

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    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    regeneration scheme along the drift direction... Boundary Lemma, which combines a deterministic Mayer–Vietoris estimate with a geometric bound relating integrated Betti numbers to sausage area via the coarea formula

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.PR 2026-04 unverdicted novelty 6.0

    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.

Reference graph

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