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Local times of Brownian motion

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dc.contributor.advisor Fouché, Willem Louw
dc.contributor.advisor Potgieter, P. H. Mukeru, Safari 2010-11-10T07:24:23Z 2010-11-10T07:24:23Z 2010-09
dc.identifier.citation Mukeru, Safari (2010) Local times of Brownian motion, University of South Africa, Pretoria, <> en
dc.description.abstract 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. en
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 PhD. (Operations Research)

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