Loewner Theory for Bernstein functions I: evolution families and differential equations

TL;DR

This paper extends Loewner theory to Bernstein functions, characterizing evolution families and their generators, with applications to inhomogeneous branching processes and complex differential equations.

Contribution

It introduces a complex-analytic framework for evolution families of Bernstein functions and characterizes their generators, advancing the understanding of inhomogeneous Markov processes.

Findings

Characterization of Herglotz vector fields generating Bernstein function evolution families

A complex-analytic proof of Silverstein's representation formula

Conditions for holomorphic self-maps to satisfy Loewner-Kufarev differential equations

Abstract

One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous Markov processes. A suitable analogue of one-parameter semigroups in the inhomogeneous setting is the notion of a (reverse) evolution family. In this paper we study evolution families formed by Bernstein functions, which play the role of Laplace exponents for inhomogeneous continuous-state branching processes. In particular, we characterize all Herglotz vector fields that generate such evolution families and give a complex-analytic proof of a qualitative description equivalent to Silverstein's representation formula for the infinitesimal generators of one-parameter semigroups of Bernstein functions. We also establish several sufficient conditions for families of holomorphic self-maps, satisfying the algebraic part in the definition of an evolution family, to be absolutely continuous and hence to be described as solutions to the generalized Loewner - Kufarev differential equation. Most of these results are then applied in the sequel paper [https://doi.org/10.48550/arXiv.2211.12442] to study continuous-state branching processes.