Speaker: Moritz Kassmann

Title: The parabolic Harnack inequality for nonlocal operators

Abstract: The parabolic Harnack inequality for nonlocal operators is an important research topic in the field of stochastic analysis, potential theory and analysis. I review the history of the last 20 years and compare the approaches of stochastic analysis and partial differential equation techniques. I then present the results of two recent papers, which were written in collaboration with Marvin Weidner. In [1] we develop a purely analytical approach for the parabolic Harnack inequality under optimal conditions for the solutions. In [2] we show why the Harnack inequality for kinetic equations such as the fractional Kolmogorov equation fails by presenting a counterexample. In the talk, we will see how both topics are closely related and influenced by results of Prof. Zhen-Qing Chen.

[1] M. Kassmann and M. Weidner, The parabolic Harnack inequality for nonlocal equations, to appear in Duke Math. J., see also arXiv:2303.05975

[2] M. Kassmann and M. Weidner, The Harnack inequality fails for nonlocal kinetic equations, arXiv:2405.05223