On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators

The paper contains a representation formula for positive solutions of linear degenerate second-order equations of the form of "sum of squares of vector fields plus a drift term" where the vector fields X_j's satisfy the Hörmander condition. It is assumed that X_j's are invariant under left translations of a Lie group and the corresponding paths satisfy a local admissibility criterion. The representation formula is established by an analytic approach based on Choquet theory. As a consequence we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.