For planar Brownian motion with nonzero drift the smoothed persistence functional counting holes in the Wiener sausage satisfies a law of large numbers: it grows linearly in time at a deterministic intensity almost surely and in L1.
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Persistence of the Wiener Sausage: Sampling Stability and a Law of Large Numbers for Drifted Planar Brownian Motion DRAFT -CURRENTLY UNDER REVIEW
For planar Brownian motion with nonzero drift the smoothed persistence functional counting holes in the Wiener sausage satisfies a law of large numbers: it grows linearly in time at a deterministic intensity almost surely and in L1.