#
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.

Self-controlled

Whitney decomposition

minimal thinness

Beurling slow varying