On parabolicity and area growth of minimal surfaces

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.