*Mathematical Methods in the Applied Sciences*Internal Bernstein Functions and Lévy-Laplace exponents

28 Feb 2020

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11 Jun 2020

11 Jun 2020

11 Jun 2020

11 Jun 2020

Bertoin, Roynette et Yor \cite{bertion} described new
connections between the class $\Bd$ of
L\’evy-Laplace exponents $\Psi$ (also
called the class (sub)critical branching mechanism) and the class of
Bernstein functions ($\BF$) which are internal, i.e.
those Bernstein functions $\phi$ s.t.
$\Psi \circ \phi$ remains
a Bernstein function for every $\Psi$. We complete
their work and illustrate how the class f internal function is rich from
the stochastic point of view. It is well known that every
$\phi \in \BF$
corresponds univocally to: (i) a subordinator
${(X_t)}_{t\geq 0}$ (or equivalently to
transition semigroups
${\big(\pr(X_t\in
dx)\big)}_{t\geq 0}$; (ii) a
L\’evy measure $\mu$ (which controls the
jumps of the subordinator). It is also known that, on
$\oi$, the measure $\pr(X_t
\in dx)/t$ converges vaguely to $\dd
\delta_0(dx)+ \mu(dx)$ as
$t\to 0$, where $\dd$ is the drift
term, but rare are the situations where we can compare the transition
semigroups with the L\’evy measure. Our extensive
investigations on the composition of L\’evy-Laplace
exponents $\Psi$ with Bernstein functions show, for
instance, this remarkable facts: $\phi$ is internal is
equivalent to: (a) $\phi^2 \in
\BF$ or to (b) $t\mu(dx) -
\pr(X_t\in dx)$ is a positive measure on
$\oi$. We also provide conditions on
$\mu$ insuring that $\phi$ is
internal. We also show L\’evy-Laplace exponents are
closely connected to the class of Thorin Bernstein function and provide
conditions on $\mu$ insuring that
$\phi$ is internal.