We study the persistent homology of the offset filtration generated by the range of a planar Brownian motion with constant nonzero drift. The members of this filtration are the Wiener sausages of increasing radius, and the degree-one persistence diagram records the birth and death of holes in the thickened trace as the radius varies. Our first result is a sampling theorem: for any continuous path in R d observed on a time grid $π$n the bottleneck distance between the persistence diagram of the continuous offset filtration and that of the sampled point cloud is bounded by the pathwise modulus of continuity $ω$X (|$π$n|). For Brownian motion this yields the almost-sure rate O |$π$n| log(1/|$π$n|) . Our second and main result is a law of large numbers for the drifted planar case. For every bounded Borel weight $ψ$ supported on a compact radius window [r0, r1] with r0 > 0, the smoothed persistence functional $Φ$ $ψ$ (T ), where $β$ T 1 (r) counts the holes in the radius-r sausage at time T , satisfies $Φ$ $ψ$ (T )/T $\rightarrow$ $ρ$ $ψ$ almost surely and in L 1 for a deterministic constant $ρ$ $ψ$ . This yields a finite positive intensity measure on the radius axis that governs the linear growth of topological complexity. The proof introduces a regeneration scheme along the drift direction: projecting the planar path onto the drift axis produces a one-dimensional Brownian motion with positive drift, whose ladder hits and bounded-backtracking events generate i.i.d. path blocks. The non-additivity of topology under concatenation is controlled by a Boundary Lemma, which combines a deterministic Mayer-Vietoris estimate with a geometric bound relating integrated Betti numbers to sausage area via the coarea formula. A Betti-curve representation converts the two-parameter persistence problem into a one-parameter family of fixed-radius hole counts, making the regeneration argument possible.