  We establish parabolicity and quadratic area growth for minimal
surfaces-with-boundary contained in regions of R^3 which are within a
sub-logarithmic factor of the exterior of a cone. Unlike previous work showing
that these two properties hold for minimal surfaces-with-boundary contained
between two catenoids, we do not make use of universal superharmonic functions.
Instead, we use stochastic methods, which have the additional feature of giving
a type of parabolicity in a more general context than Brownian motion on a
minimal surface.
