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 ${\bf e}_α: x\to x^α$ where $x\geq0$ and $0<α\leq 1/2$, and in particular the critical parameter $α=1/2$, play a distinguished role.