Regular points and the boundary-value problem for a hypoelliptic process in a general smooth domain

We fully describe the set of regular and irregular boundary points for a general hypoelliptic stochastic process $(X(t){)}_{t\ge 0}$ absorbed outside a $C^{3,1}$ domain $\mathrm{\Omega }$. The well-posedness of the associated boundary-value problem on $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$. In particular, characteristic and non-characteristic points may coexist on $\mathrm{\partial }\mathrm{\Omega }$, 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 $\mathrm{\partial }\mathrm{\Omega }$. In certain cases involving a $C^{\mathrm{\infty }}$ domain $\mathrm{\Omega }$ and $C^{\mathrm{\infty }}$ coefficients, the discontinuity set is even of positive Lebesgue surface measure on $\mathrm{\partial }\mathrm{\Omega }$. Finally, we highlight a cone condition under which the continuity is preserved at the boundary.