Internal Bernstein functions and Lévy‐Laplace exponents

Abstract

Bertoin, Roynette and Yor (2004) described new connections between the class of Lévy‐Laplace exponents Ψ (also called the class of (sub)critical branching mechanism) and the class of Bernstein functions ( ) which are internal, that is, those Bernstein functions ϕ s.t. Ψ∘ϕ remains a Bernstein function for every Ψ. We complete their work and illustrate how the class of internal function is rich from the stochastic point of view. It is well known that every corresponds to (i) a subordinator (Xt)t ≥ 0 (or equivalently to transition semigroups and (ii) a Lévy measure μ (which controls the jumps of the subordinator). It is also known that, on (0,∞), the measure converges vaguely to dδ0(dx)+μ(dx) as t→0, where d is the drift term, but rare are the situations where we can compare the transition semigroups with the Lévy measure. Our extensive investigations on the composition of Lévy‐Laplace exponents Ψ with Bernstein functions show, for instance, these remarkable facts: ϕ is internal is equivalent to (a) or to (b) is a positive measure on (0,∞). We also provide conditions on μ insuring internality for ϕ and illustrate how Lévy‐Laplace exponents are closely connected to the class of Thorin Bernstein function.