Prof. Giulio Tralli “Operators with linear drift terms with imaginary spectrum, and their Liouville properties”

In this talk we discuss the validity of Liouville theorems for non-negative solutions to hypoelliptic equations with constant diffusion and linear drift. Assuming that the drift term has imaginary spectrum, we show such one-sided Liouville property as a by-product of an invariant Harnack inequality for ancient solutions to parabolic equations of Kolmogorov type.
We focus on the role of the large-scale geometry associated to the drift: the relevance of the hypoelliptic class under discussion (even for the case of elliptic operators) appears naturally in our approach.

The talk is based on a joint work with A.E. Kogoj and E. Lanconelli.