Persistent Homology of the Planar Wiener Sausage: Brownian Scaling and a Logarithmic Expectation Law

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 [a,b]⊂(0,∞), we consider the smoothed Betti-curve observable [r₀,r₁] Φ_ψ(T) = ∫r₀r₁ β₁T ( r ) ψ( r ) dr. We prove that there exist absolute constants 0