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

Abstract

This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $${\partial }_{t}u(x,t)=\sum _{j=1}^m{X}_j^{2}u(x,t)+X_{0}u(x,t)(x,t)\in {\mathbb{R}}^N\times ]-\infty ,T[,$$ proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.

The results of the present version recover most of the ones in the previous version, but, on top of it, this new version contains some further new and interesting results

References in corpus (2)

• Poincare inequality and the uniqueness of solutions for the heat equation associated with subelliptic diffusion operators
• Harnack inequality for hypoelliptic second order partial differential operators