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

Boundary behavior of positive solutions of the heat equation on a stratified Lie group