Persistent Homology of the Wiener Sausage II: A Central Limit Theorem for Drifted Planar Brownian Motion

Abstract

Let $X_t=B_t+\mu t$, $t\ge 0$, be planar Brownian motion with nonzero drift, and let $K_t^r=\{x\in {\mathbb{R}}^2:\mathrm{dist}(x,X[0,t])\le r\}$ be the radius-$r$ Wiener sausage up to time $t$. For a bounded Borel function $\psi$ supported in a compact interval $[r_0,r_1]\subset (0,\infty )$, consider the smoothed Betti-curve functional ${\Phi }_{\psi }(t):={\int }_{r_0}^{r_1}{\beta }_1^t(r)\psi (r)dr$, where ${\beta }_1^t(r)$ denotes the number of holes of $K_t^r$. In a previous paper, a regeneration scheme along the drift direction was used to prove a law of large numbers for ${\Phi }_{\psi }(t)$. In the present paper we prove the corresponding central limit theorem. More precisely, there exist a deterministic constant ${\rho }_{\psi }$ and a variance ${\sigma }_{\psi }^2\ge 0$ such that $({\Phi }_{\psi }(t)-{\rho }_{\psi }t)/\sqrt{t}{\stackrel{d}{\to }}_{t\to \infty }\mathcal{N}(0,{\sigma }_{\psi }^2)$. We also obtain the finite-dimensional Gaussian limit for finitely many test functions. The proof preserves the regenerative structure of the law of large numbers, but requires a new $L^2$ analysis of the topological interface terms created at regeneration cuts. The key input is a finite-time polynomial moment bound for integrated hole counts of the Wiener sausage. This yields square-integrability of cycle increments, within-cycle oscillations, and the last incomplete-cycle remainder, which in turn allows one to combine a standard central limit theorem for stationary $1$-dependent sequences with a renewal time-change argument.