*preprint*

**Inserted:** 21 jul 2022

**Last Updated:** 21 jul 2022

**Year:** 2021

**Abstract:**

In the framework of the Potential Theory we prove existence and uniqueness for the Perron-Wiener-Brelot solution to the Dirichlet problem related to a family of totally degenerate, in the sense of Bony, differential operators. We also state and prove a Wiener-type criterium and an exterior cone condition for boundary regularity. Our results apply to a wide family of strongly degenerate operators that includes the following example $\mathcal{L} = t^2\Delta_x + \langle x, \nabla_y \rangle -\partial_t$, for $(x,y,t) \in \mathbb{R}^N \times \mathbb{R}^{N} \times \mathbb{R}$.