Let Xₜ = Bₜ + μt, t 0, be planar Brownian motion with nonzero drift, and let Kₜʳ = \x R²: { dist (x, X0, 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): = ₑ䃐^r₁ 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 dₓ 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.
Tristan Guillaume (Wed,) studied this question.