Minimal Fine Derivatives and Brownian Excursions

Abstract

Let f be an analytic function defined on D [is a subset of] [complex numbers] C. If [the derivative of the function f at the point x] has a limit when [the set] x [into the set] z [is an element of the set partial derivative] D in the minimal fine topology then the limit will be called a minimal fine derivative. Several results concerning the existence of such derivatives are given. The relationship between minimal fine derivatives and angular derivatives is studied. An application to Brownian excursions is presented.