Regular points and the boundary-value problem for a hypoelliptic process in a general smooth domain
Abstract
We fully describe the set of regular and irregular boundary points for a general hypoelliptic stochastic process (X(t))t≥0 absorbed outside a C3, 1 domain Ω. The well-posedness of the associated boundary-value problem on Ω is also established in the classical sense. Let us emphasize that, throughout this work, no assumption is made regarding the inherent structure of the boundary of Ω. In particular, characteristic and non-characteristic points may coexist on ∂Ω, thus extending previous results obtained in the literature. We show that, unlike previous settings, the solution to this boundary-value problem may exhibit discontinuity at the boundary ∂Ω. In certain cases involving a C∞ domain Ω and C∞ coefficients, the discontinuity set is even of positive Lebesgue surface measure on ∂Ω. Finally, we highlight a cone condition under which the continuity is preserved at the boundary.
Related
- The Dirichlet problem with boundary data of fractional smoothness in domains with Ahlfors regular boundary
- Fundamentals of Potential theory for a hypoelliptic process
- Derivation of the General Solution of the Black-Scholes Boundary-Value Problem
- Regular boundary conditions and analytic K-homology
- Discreteness of the Steklov Spectrum for Exterior Non-Compact Free-Boundary Minimal Surfaces with Regular Ends of Finite Total Curvature
BibTeX
@misc{ramil2026regular,
title = {{Regular points and the boundary-value problem for a hypoelliptic process in a general smooth domain}},
author = {Mouad Ramil and Mathias Rousset},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39409},
eprint = {2609.39409},
archivePrefix = {arXiv},
url = {https://inria.hal.science/hal-05773317v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regular points and the boundary-value problem for a hypoelliptic process in a general smooth domain
AU - Mouad Ramil
AU - Mathias Rousset
PY - 2026
DA - 2026/09/30
PB - HAL
DO - 10.48550/arxiv.2609.39409
UR - https://inria.hal.science/hal-05773317v1
AB - We fully describe the set of regular and irregular boundary points for a general hypoelliptic stochastic process (X(t))_t≥0 absorbed outside a C^3, 1 domain Ω. The well-posedness of the associated boundary-value problem on Ω is also established in the classical sense. Let us emphasize that, throughout this work, no assumption is made regarding the inherent structure of the boundary of Ω. In particular, characteristic and non-characteristic points may coexist on ∂Ω, thus extending previous results obtained in the literature. We show that, unlike previous settings, the solution to this boundary-value problem may exhibit discontinuity at the boundary ∂Ω. In certain cases involving a C^∞ domain Ω and C^∞ coefficients, the discontinuity set is even of positive Lebesgue surface measure on ∂Ω. Finally, we highlight a cone condition under which the continuity is preserved at the boundary.
ER -