Kolmogorov-type operators and Krylov-Safonov theory: state of the art and partial results

In this talk we will discuss the validity of Harnack inequalities for linear evolution equations modelled after the Kolmogorov operator. The main focus will be on nondivergence form equations with non-smooth coefficients, and on the absence of an analogue of the ABP maximum principle and of the Krylov-Safonov theory in this setting. In particular, we will highlight as the (very general) crucial ingredient some quantitative point-to-measure estimate for nonnegative subsolutions. With this perspective in mind, we will show a potential theory approach (established in a 2019 joint work with F. Abedin) which allows to prove invariant Harnack inequalities for Kolmogorov-type operators with coefficients satisfying either a Cordes-Landis assumption or a continuity hypothesis.