| Abstract | Cilj ovog rada je bio istražiti obilježja trajektorija Brownova gibanja pomoću tehnika koje koriste Hausdorffovu dimenziju. Pokazali smo da Brownovo gibanje gotovo sigurno nisu nigdje diferencijabilne te da je za \(\alpha < 1/2\) svugdje lokalno \(\alpha\)-Hölder neprekidno. Jedno od zanimljivih svojstava planarnog Brownova gibanja je to da se Brownovo gibanje s vjerojatnošću jedan vraća u okolinu iz koje je krenulo, odnosno da svaku okolinu posjećuje beskonačno mnogo puta. Spomenuli smo i važnu klasu slučajnih procesa - Markovljeve procese. U radu smo također promatrali razne skupove izvedene iz Brownova gibanja poput skupa nultočki, skupa lokalnih maksimuma i skupa rekorda, pri čemu se ključnim pokazala činjenica da Brownovo gibanje zadovoljava jako Markovljevo svojstvo. Rezultati koje smo dobili pokazuju da, iako su trajektorije Brownova gibanja gotovo sigurno neprekidne, skup nultočki je beskonačan i bez izoliranih točaka. Osim toga, u slučaju linearnoga Brownova gibanja vidjeli smo da je to, kao i skup rekorda, fraktalni skup Hausdorffove dimenzije 1/2. Također, pokazali smo da je dimenzija grafa linearnoga Brownova gibanja jednaka 3/2, a u slučaju višedimenzionalnoga Brownova gibanja dimenzije grafa i slike jednake su i iznose 2. Na kraju rada dokazali smo dva klasična teorema iz područja koje se bavi Hausdorffovom dimenzijom, Frostmanovu lemu i McKeanov teorem. McKeanov teorem pokazuje da je dimenzija skupa na koji se preslikava Brownovo gibanje dva puta veća od dimenzije početnog skupa. |
| Abstract (english) | 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. |
