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

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