Loewner Theory for Bernstein functions I: evolution families and 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.
Keywords
Cite
@article{arxiv.2206.04753,
title = {Loewner Theory for Bernstein functions I: evolution families and differential equations},
author = {Pavel Gumenyuk and Takahiro Hasebe and José-Luis Pérez},
journal= {arXiv preprint arXiv:2206.04753},
year = {2023}
}
Comments
A reference to the sequel preprint "Loewner Theory for Bernstein functions II" is added in the abstract. Ver.2: References [19] and [32] have been updated and a small error on page 22 (the very end of the proof of Theorem 3) has been corrected Ver.3: the grant info has been updated