Nonlocal parabolic De Giorgi classes

We study the local behavior of the elements of a specific energy class of functions, called the nonlocal parabolic ($p$-homogenous) De Giorgi class. First we carry on an analysis of their local boundedness under optimal tail conditions, and then prove several weak Harnack inequalities, measure theoretical propagation lemmas, and a parabolic Harnack inequality. We show a full proof of the local H\"{o}lder continuity, eventually establishing a Liouville-type rigidity property. Finally, as an application of our method, we prove a state-of-the-art nonlocal Harnack inequality for nonnegative solutions of the nonlocal Trudinger equation.