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

Abstract

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 $K_T=B\left(\left[0,T\right]\right)$, 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 $β_1^T\left(r\right)$ is the number of bounded complementary components of $K_T^{\left(r\right)}$. For a bounded nonnegative Borel function $ψ$ supported in a compact interval $\left[a,b\right]\subset\left(0,\infty\right)$, we consider the smoothed Betti-curve observable $\left[r_0,r_1\right] \mathrmΦ_ψ\left(T\right) = \int_{r_0}^{r_1} β_1^T \left( r \right) ψ\left( r \right) dr$. We prove that there exist absolute constants 0