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

This paper establishes necessary and sufficient condition for the regularity of a characteristic top boundary point of an arbitrary open subset of[inline-graphic not available: see fulltext] ([inline-graphic not available: see fulltext]) for the diffusion (or heat) equation. The result implies asymptotic probability law for the standard[inline-graphic not available: see fulltext]-dimensional Brownian motion.