Loewner Theory for Bernstein functions II: applications to inhomogeneous continuous-state branching processes

TL;DR

This paper applies Loewner Theory to characterize time-inhomogeneous continuous-state branching processes using Bernstein functions, establishing ODE representations and conditions for embeddability, with implications for both continuous and discrete state spaces.

Contribution

It introduces a novel application of Loewner Theory to inhomogeneous branching processes, providing new characterizations and ODE frameworks for their Laplace exponents.

Findings

Laplace exponents form topological reverse evolution families of Bernstein functions.

Established a Loewner-Kufarev type ODE for Laplace exponents under regularity conditions.

Derived conditions for embedding discrete-state branching processes into continuous-state processes.

Abstract

This paper continues the research project launched in [Constr. Approx. (2025) https://doi.org/10.1007/s00365-023-09675-9] and aimed at studying time-inhomogeneous one-dimensional branching processes (mainly on a continuous but also on a discrete state space) with the help of recent achievements in Loewner Theory dealing with evolution families of holomorphic self-maps in simply connected domains of the complex plane. Under a suitable stochastic continuity condition, we show that the families of the Laplace exponents of branching processes on$~[0,\infty]$ can be characterized as topological (i.e. depending continuously on the time parameters) reverse evolution families whose elements are Bernstein functions. For the case of a stronger regularity w.r.t. time, we establish a Loewner-Kufarev type ODE for the Laplace exponents and characterize branching processes with finite mean in terms of the vector field driving this ODE. Similar results are obtained for families of probability generating functions of branching processes on the discrete state space $\{0,1,2,\ldots\}\cup\{\infty\}$. In addition, we find a necessary and sufficient condition for "spatial" embeddability of such branching processes into branching processes on$~[0,\infty]$. Finally, we give some probabilistic interpretations of the Denjoy-Wolff point at$~0$ and at$~\infty$.