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

Tristan Guillaume

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 $π$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 $ω$X (|$π$n|). For Brownian motion this yields the almost-sure rate O |$π$n| log(1/|$π$n|) . Our second and main result is a law of large numbers for the drifted planar case. For every bounded Borel weight $ψ$ supported on a compact radius window [r0, r1] with r0 > 0, the smoothed persistence functional $Φ$ $ψ$ (T ), where $β$ T 1 (r) counts the holes in the radius-r sausage at time T , satisfies $Φ$ $ψ$ (T )/T $\rightarrow$ $ρ$ $ψ$ almost surely and in L 1 for a deterministic constant $ρ$ $ψ$ . 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.

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

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 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 X (|n|). For Brownian motion this yields the almost-sure rate O |n| log(1/|n|) . Our second and main result is a law of large numbers for the drifted planar case. For every bounded Borel weight supported on a compact radius window [r0, r1] with r0 > 0, the smoothed persistence functional (T ), where T 1 (r) counts the holes in the radius-r sausage at time T , satisfies (T )/T almost surely and in L 1 for a deterministic constant . 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.

Paper Structure

This paper contains 29 sections, 35 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Setup and preliminaries
  3. Offsets of compact sets and Wiener sausage filtrations
  4. Persistence diagrams and their metrics
  5. q-tameness of offset filtrations
  6. Stability with respect to Hausdorff perturbations
  7. Persistence counting measures, Betti curves, and smoothed functionals
  8. Sampling stability for continuous paths
  9. Continuous and sampled offset filtrations
  10. Interleaving of offset filtrations
  11. The sampling theorem
  12. Brownian motion: the Lévy-modulus rate
  13. Diffusions with globally Lipschitz coefficients
  14. Practical variants and computational remarks
  15. Regeneration structure for drifted planar Brownian motion
  16. ...and 14 more sections

Key Result

proposition 1

Let $A \subset \mathbb{R}^{d}$ be compact. Then the offset filtration $\mathcal{F}(A) = (A^{(r)})_{r \geq 0}$ is $q$-tame for every $q \geq 0$. Consequently, the persistence diagram ${Dgm}_{q}(A)$ is well defined. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (70)

  • proposition 1
  • proof
  • theorem 1: stability for offsets
  • proof
  • proposition 2: Hausdorff control by the modulus of continuity
  • proof
  • lemma 1: offset interleaving
  • proof
  • corollary 1
  • proof
  • ...and 60 more