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 $\sqrt{T}$. 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 $K_T=B\left([0,T]\right)$, and let $K_T^{\left(r\right)}$ be the radius-$r$ Wiener sausage. Since $K_T^{\left(r\right)}$ is connected, its first Betti number ${\beta }_1^T\left(r\right)$ is the number of bounded complementary components of $K_T^{\left(r\right)}$. For a bounded nonnegative Borel function $\psi$ supported in a compact interval $[a,b]\subset (0,\mathrm{\infty })$, we consider the smoothed Betti-curve observable $[r_0,r_1]{\mathrm{\Phi }}_{\psi }\left(T\right)={\int }_{r_0}^{r_1}{\beta }_1^T\left(r\right)\psi \left(r\right)dr$. We prove that there exist absolute constants 0