Fundamentals of Potential theory for a hypoelliptic process

This article aims at solving open questions related to the well-posedness of Potential theory objects in a hypoelliptic setting. We provide here the expression of the equilibrium measure, capacity and electrostatic energy between smooth open sets $\text{A}$ and $\text{B}$ for the underdamped Langevin process. Additionally, the capacity is shown to be the solution of two variational principles which extend the well-known Dirichlet and Thomson principles in the hypoelliptic setting. Due to the degeneracy of the operator at hand, the results display significant differences with the standard elliptic setting.