  In this article, we are concerned with a certain type of boundary behavior of
positive solutions of the heat equation on a stratified Lie group at a given
boundary point. We prove that a necessary and sufficient condition for the
existence of the parabolic limit of a positive solution $u$ at a point on the
boundary is the existence of the strong derivative of the boundary measure of
$u$ at that point. Moreover, the parabolic limit and the strong derivative are
equal. We also construct an example of a positive measure on the Heisenberg
group to show that the set of all points where strong derivative exists is
strictly larger than the set of Lebesgue points of the measure.
