On parabolic convergence of positive solutions of the heat equation

In this article, we study certain type of boundary behaviour of positive solutions of the heat equation on the upper half-space of ${\mathbb{R}}^{n+1}$. We prove that the existence of the parabolic limit of a positive solution of the heat equation at a point in the boundary is equivalent to the existence of the strong derivative of the boundary measure of the solution at that point. Moreover, the parabolic limit and the strong derivative are equal.