Abstract :
[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.
