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PreprintCC BY 4.0HAL

Fundamentals of Potential theory for a hypoelliptic process

  • Mouad Ramil
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Abstract

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 A and 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.

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Cite

BibTeXRIS
BibTeX
@misc{ramil2026fundamentals,
  title = {{Fundamentals of Potential theory for a hypoelliptic process}},
  author = {Mouad Ramil},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21781},
  eprint = {2610.21781},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05802116v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Fundamentals of Potential theory for a hypoelliptic process
AU  - Mouad Ramil
PY  - 2026
DA  - 2026/10/15
PB  - HAL
DO  - 10.48550/arxiv.2610.21781
UR  - https://hal.science/hal-05802116v1
AB  - 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 A and 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.
ER  -