| Reference : FRACTAL DIMENSION AND POINT-WISE PROPERTIES OF TRAJECTORIES OF FRACTIONAL PROCESSES |
| Dissertations and theses : Doctoral thesis | |||
| Physical, chemical, mathematical & earth Sciences : Mathematics | |||
| http://hdl.handle.net/10993/51093 | |||
| FRACTAL DIMENSION AND POINT-WISE PROPERTIES OF TRAJECTORIES OF FRACTIONAL PROCESSES | |
| English | |
| [fr] Dimension fractale et propriétés ponctuelles des trajectoires des processus fractionnaires | |
Daw, Lara  [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >] | |
| 20-May-2022 | |
| Lara Daw, Esch-sur-Alzette, Luxembourg | |
| DOCTEUR DE L’UNIVERSITE DU LUXEMBOURG EN MATHEMATIQUES | |
| 169 | |
| Nourdin, Ivan | |
| Seuret, Stephane | |
| [en] Rosenblatt process ; Fractional Brownian motion ; Image set ; Level set ; Sojourn times ; Wavelet series ; Slow/Ordinary/Rapid points ; Fractal dimensions ; Macroscopic Hausdorff dimension ; Potential theory for dimensions ; Projection theorem | |
| [en] The topics of this thesis lie at the interference of probability theory with dimensional
and harmonic analysis, accentuating the geometric properties of random paths of Gaussian and non-Gaussian stochastic processes. Such line of research has been rapidly growing in past years, paying off clear local and global properties for random paths associated to various stochastic processes such as Brownian and fractional Brownian motion. In this thesis, we start by studying the level sets associated to fractional Brownian motion using the macroscopic Hausdorff dimension. Then as a preliminary step, we establish some technical points regarding the distribution of the Rosenblatt process for the purpose of studying various geometric properties of its random paths. First, we obtain results concerning the Hausdorff (both classical and macroscopic), packing and intermediate dimensions, and the logarithmic and pixel densities of the image, level and sojourn time sets associated with sample paths of the Rosenblatt process. Second, we study the pointwise regularity of the generalized Rosenblatt and prove the existence of three kinds of local behavior: slow, ordinary and rapid points. In the last chapter, we illustrate several methods to estimate the macroscopic Hausdorff dimension, which played a key role in our results. In particular, we build the potential theoretical methods. Then, relying on this, we show that the macroscopic Hausdorff dimension of the projection of a set E ⊂ R^2 onto almost all straight lines passing through the origin in R^2 depends only on E, that is, they are almost surely independent of the choice of straight line. | |
| Researchers ; Professionals ; Students | |
| http://hdl.handle.net/10993/51093 |
| File(s) associated to this reference | ||||||||||||||
|
Fulltext file(s):
| ||||||||||||||
All documents in ORBilu are protected by a user license.
