Regular Boundary Points and Exit Distributions for Parabolic Differential Operators

Abstract

We show that the regularity of a boundary point for a parabolic differential operator in divergence form is under some geometric assumptions equivalent to the property that the density of the exit distribution for a time reversed process vanishes at that point. We give regularity and irregularity criterions for equations with variable coefficients. Thus, the known result on the Fulks measure that states that the density with respect to the Lebesgue measure vanishes at the point opposite to the center of the heat ball (see Fulks (Proc. Am. Math. Soc. 17, 6–11 1966)) can be extended to exit distributions for more general regions and parabolic differential operators.