2022/06/09 by Pavel Gumenyuk, Takahiro Hasebe, Gumenyuk, Pavel +3
Mathematics · #30C80 #30D05 #37F44 #60J80 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals #Nonlinear Differential Equations Analysis #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2206.04753
openalex publication_date 2022/06/09 · openalex created_date 2022/06/14 · openalex updated_date 2026/07/28
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.