  This note contains a representation formula for positive solutions of linear
degenerate second-order equations of the form $$ \partial_t u (x,t) =
\sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \qquad (x,t) \in \mathbb{R}^N \times\,
]- \infty ,T[,$$ 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.
