Relations between cone-parameter Lévy processes and convolution semigroups

Abstract

Cone-parameter Lévy processes and convolution semigroups on $R^d$ are defined. Here, cone-parameter Lévy processes have stationary independent increments along increasing sequences on the cone. This property ensures that subordination of a cone-parameter Lévy process by an independent cone-valued cone-parameter Lévy process yields a new cone-parameter Lévy process. It is shown that a cone-parameter Lévy process induces a cone-parameter convolution semigroup. The converse statement, that any convolution semigroup appears in this way, is however not true. In particular we show that there is no Brownian motion with parameter in the set of nonnegative-definite symmetric $d\times d$ matrices. The question when a given cone-parameter convolution semigroup is generated by a Lévy process is studied. It is shown that this is the case if one of the following three conditions is satisfied: $d=1$; the convolution semigroup is purely non-Gaussian; or $K$ is isomorphic to $R_{+}^N$.