Studying the geometric properties of the sample paths of a fractional Brown- ian motion $(B_t)_{t\geq 0}$ has many approaches. One of them is describing the (asymptotic) proportion of time spent by the process in a given region which is known by the notion of sojourn times. Sojourn times have been studied by many authors and play a key role in understanding various features of the paths of stochastic processes, especially those of fractional Brownian motion.
After defining what is a fractional Brownian motion and introducing some no- tions related to it such as sojourn sets, I will introduce the microscopic Hausdorff dimension and two different densities. Computing the macroscopic Hausdorff dimension, logarithmic and macroscopic densities of the sojourn sets will give us some geometric properties of the path of fractional Brownian motion. I will also give a uniform macroscopic dimension result for the fractional Brownian level sets.
Keywords: Sojourn time; logarithmic density; pixel density; macroscopic Hausdorff dimension; fractional Brownian motion.
