We study degree-one persistent homology of the planar Wiener-sausage filtration generated by standard Brownian motion without drift. In the drifted case, regeneration along the drift direction leads to linear-in-time laws for persistent-homological observables. In the recurrent zero-drift case, this renewal structure disappears. The organizing mechanism is instead Brownian self-similarity: the persistence diagram at time T is equal in law to the image of the unit-time diagram under spatial dilation by T. Consequently, large-time questions on fixed radius windows are transformed into small-radius questions for the unit-time Brownian trace. Let B be standard planar Brownian motion, let KT=B (0, T), and let KT^ (r) be the radius-r Wiener sausage. Since KT^ (r) is connected, its first Betti number β₁T (r) is the number of bounded complementary components of KT^ (r). For a bounded nonnegative Borel function ψ supported in a compact interval, b (0, ), we consider the smoothed Betti-curve observable ₀, r₁ Φ_ψ (T) = ₑ䃐^r₁ β₁T (r) ψ (r) dr. We prove that there exist absolute constants 0
Tristan Guillaume (Mon,) studied this question.