Some Connections Between (Sub)Critical Branching Mechanisms and Bernstein Functions

Abstract

We describe some connections, via composition, between two functional spaces: the space of (sub)critical branching mechanisms and the space of Bernstein functions. The functions ${\mathbf{e}}_{\alpha }:x\to x^{\alpha }$ where $x\ge 0$ and $0<\alpha \le 1/2$, and in particular the critical parameter $\alpha =1/2$, play a distinguished role.