Minimally thin sets below a function graph

Journal article, 2004

A set $E$ is minimally thin at a boundary point, $\xi$, if the Martin kernel with pole at $\xi$ does not coincide with its balayage on $E$. %it is not ``big enough to lift the Poisson kernel''. Or in a probabilistic language: There is a non-zero probability that a Brownian motion that is conditioned to exit at $\xi$ will avoid the set $E$. We will consider a special class of sets $E$, namely sets in the upper half-space that lies between the graph of a function and the boundary of the half-space. %(so called epigraphs). Brelot and Doob gave in 1963 an integral criterion for positive non-decreasing functions for minimally thinness of $E$. In 1991 Gardiner showed that the same criterion holds for the class of Lipschitz continuous functions. We will generalize these results to the class {\em self-controlled} functions, which is similar to the {\em Beurling slow varying} class of functions.