# Brownovo gibanje i Hausdorffova dimenzija

Lazić, Petra (2015) Brownovo gibanje i Hausdorffova dimenzija. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

The aim of this thesis was to explore the nature of Brownian paths by applying techniques that calculate the Hausdorff dimension. We have shown that, almost surely, Brownian motion is nowhere differentiable and, for any $\alpha < 1/2$, everywhere locally $\alpha$-Hölder continuous. We have also seen that planar Brownian motion is neighbourhood recurrent, that is, it visits every neighbourhood in the plane infinitely often. In the thesis we have also studied various processes and sets derived from Brownian motion, such as the zero set, the set of all record times and the set of times where the local maxima are attained. Here, we made great use of the strong Markov property. The results show that, despite the fact that the sample paths of Brownian motion are almost surely continuous, the zero set is infinite and with no isolated points. Moreover, in the case of linear Brownian motion we have seen that the zero set is an example of a fractal set of Hausdorff dimension 1/2, just like the set of record times. Further, we have shown that the graph of one dimensional Brownian motion has dimension 3/2, and the graph and range of $d$-dimensional Brownian motion, for $d > 2$, both have dimension 2. Finally, proves of two classical results of the Hausdorff dimension theory were presented: Frostman’s lemma and McKean’s theorem. The result of McKean’s theorem shows that the image of a set under a Brownian motion has twice the dimension of the original set.