Some remarks on stable stochastic processes and?-superharmonic functions

Abstract

An integral representation is established for a stable multidimensional probability density. It is used for a new and direct proof of the fact that anα-superharmonic function considered on the trajectories of a stable symmetric stochastic process with parameter a is a super-martingale. It is moreover established that the stable density belongs to the convex cone generated by the functions efα(r)(x) of M. Riesz.