During the last 20 years, there was a huge interest in the study of regularity results for solutions to nonlocal equations governed by operators that are modeled upon the fractional Laplacian. In this talk, I will focus on the study of solutions to time-dependent nonlocal equations with bounded measurable coefficients. In particular, I will report on two recent results on the nonlocal parabolic Harnack inequality, highlighting several purely nonlocal phenomena, such as the failure of Hölder estimates for certain weak solutions and the validity of time-insensitive Harnack estimates.

This talk is based on several joint works with Moritz Kassmann and Naian Liao.