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Abstract:
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After a review of the notions of Hausdorff and Fourier dimensions from fractal geometry
and Fourier analysis and the properties of local times of Brownian motion, we study the
Fourier structure of Brownian level sets. We show that if δa(X) is the Dirac measure
of one-dimensional Brownian motion X at the level a, that is the measure defined by
the Brownian local time La at level a, and μ is its restriction to the random interval
[0, L−1
a (1)], then the Fourier transform of μ is such that, with positive probability, for all
0 ≤ β < 1/2, the function u → |u|β|μ(u)|2, (u ∈ R), is bounded. This growth rate is the
best possible. Consequently, each Brownian level set, reduced to a compact interval, is
with positive probability, a Salem set of dimension 1/2. We also show that the zero set
of X reduced to the interval [0, L−1
0 (1)] is, almost surely, a Salem set. Finally, we show
that the restriction μ of δ0(X) to the deterministic interval [0, 1] is such that its Fourier
transform satisfies E (|ˆμ(u)|2) ≤ C|u|−1/2, u 6= 0 and C > 0.
Key words: Hausdorff dimension, Fourier dimension, Salem sets, Brownian motion,
local times, level sets, Fourier transform, inverse local times. |
