## Transport phenomena in collective dynamics: from micro to social hydrodynamics

### The parabolic fractional obstacle problem

Alessio Figalli

ETH Zürich

Abstract:

The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump Levy processes. Most of the results in the latest years dealt with the stationary case, where the presence of frequency-type formulas allows one to study the fine structure of the free boundary. In this talk I'll give an overview of the problem and I'll present new recent results about the regularity of the solution and the free boundary in the full parabolic setting, where no frequency formulas are available.