Probability theory uses mathematical models to analyze random phenomena. The focus in this project will be on a model, called fractional processes, which can be thought of as a kind of random fractals. Such models are often used to describe phenomena which have a very irregular behavior over time, as for example, seen in stock prices. The challenge in the project is to give a precise mathematical description of how fractional processes behave over time. More specifically, the goal is to develop methods to determine the degree of irregularity of these processes. This requires new tools and ideas, as the techniques used to analyze more classic models, breaks down in this set-up. The project runs for a four year period from February 1, 2015 to January 31, 2019, and is funded by a Sapere Aude: DFF-Starting Grant awarded by the Danish Council for Independent Research.