We show that if the transition operator \( \mathbb P_{\;\!\!\text {t}}\) of a continuous time-homogeneous (strong) Markov process X in \( I\!\!R^n \) maps infinitely differentiable functions with compact support to twice continuously differentiable functions for \( t\! >\! 0 \) , then the action of the infinitesimal operator \( \mathbb {A} \) of X on its domain coincides with the action of the differential operator \( \mathbb {D} \) of X understood in the sense of Schwartz distributions. Applying this fact to the process X stopped at the first exit time from a given open subset C of \( I\!\!R^n \) , we derive that the solution to Dynkin’s characteristic operator equation of X arising from the Dirichlet problem on C can be viewed as a weak solution to the differential operator equation of X in the sense of Schwartz distributions. A useful consequence of this identification is that the solution is infinitely differentiable on C whenever the drift and diffusion coefficients are infinitely differentiable and the differential operator \( \mathbb D\) of X is hypoelliptic (e.g. when \( \mathbb D\) satisfies the Hörmander condition). In particular, this is satisfied when X is a degenerate diffusion process in \( I\!\!R^n \) (in the sense that \( \mathbb D\) is a degenerate parabolic operator) so that the analytic existence results for parabolic PDEs are generally not available. The arguments and results extend to cover more general boundary value problems of this kind including the initial value problems as well.
