Multidimensional Kolmogorov-Petrovsky test for the boundary regularity and irregularity of solutions to the heat equation

Abstract

This paper establishes necessary and sufficient condition for the regularity of a characteristic top boundary point of an arbitrary open subset of ℝN+1 (N≥2) for the diffusion (or heat) equation. The result implies asymptotic probability law for the standard N-dimensional Brownian motion.