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# Local times of Brownian motion

 dc.contributor.advisor Fouché, Willem Louw dc.contributor.advisor Potgieter, P. H. dc.contributor.author Mukeru, Safari dc.date.accessioned 2010-11-10T07:24:23Z dc.date.available 2010-11-10T07:24:23Z dc.date.issued 2010-09 dc.identifier.citation Mukeru, Safari (2010) Local times of Brownian motion, University of South Africa, Pretoria, en dc.identifier.uri http://hdl.handle.net/10500/3781 dc.description.abstract After a review of the notions of Hausdorff and Fourier dimensions from fractal geometry en 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. dc.format.extent 1 online resource (iv, 85 leaves) dc.language.iso en en dc.subject Brownian motion en dc.subject Hausdorff dimension en dc.subject Fourier dimension en dc.subject Salem sets dc.subject Local times dc.subject Level sets dc.subject Fourier transform dc.subject Inverse local times dc.subject.ddc 519.233 dc.subject.lcsh Probabilities dc.subject.lcsh Brownian motion processes dc.subject.lcsh Fourier transformations dc.subject.lcsh Local times (Stochastic processes) dc.subject.lcsh Level set methods dc.title Local times of Brownian motion en dc.type Thesis en dc.description.department Decision Sciences dc.description.degree PhD. (Operations Research)
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