Linking fractional calculus to real data

I will review some well-known theoretical findings about fractional calculus and, in particular, the links between fractal intermittency, the Continuous Time Random Walk (CTRW) model and the emergence of Fractional Diffu- sion Equations (FDE) for anomalous diffusion. In this framework, I will show how fractional operators are associated with the existence of renewal events, a typical feature of complex systems. I will also discuss the possibile connections with critical phenomena. Then, I will introduce some statistical methods allowing to understand when a real system could be described by means of fractional models. Finally, I will show some applications to real data from nano-crystal fluores- cence intermittency, human brain dynamics and atmospheric turbulence.