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

Abstract

Let $X_t = B_t + \mu t$, $t \geq 0$, be planar Brownian motion with nonzero drift, and let $K_t^r = \{x \in \mathbb{R}^2 : {\rm dist}(x, X[0,t]) \leq r\}$ 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 $\sigma_\psi^2 \geq 0$ such that $(\Phi_\psi(t) - \rho_\psi t)/\sqrt{t} \xrightarrow{d}_{t \to \infty} \mathcal{N}(0, \sigma_\psi^2)$. 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.