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

Let Xₜ = Bₜ + μt, t ≥ 0, be planar Brownian motion with nonzero drift, and let Kₜʳ = ∈ R² : dist(x, X[0,t]) ≤ r\ be the radius-r Wiener sausage up to time t. For a bounded Borel function $ψ$ supported in a compact interval [r₀, r₁] ⊂ (0,∞), consider the smoothed Betti-curve functional Φ_ψ(t) := ∫r₀r₁ β₁ᵗ(r)\,ψ(r)\,dr, where β₁ᵗ(r) denotes the number of holes of Kₜʳ. In a previous paper, a regeneration scheme along the drift direction was used to prove a law of large numbers for Φ_ψ(t). In the present paper we prove the corresponding central limit theorem. More precisely, there exist a deterministic constant ρ_ψ and a variance σ_ψ² ≥ 0 such that (Φ_ψ(t) - ρ_ψt)/√t dt → ∞ N(0, σ_ψ²). 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 L² 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.