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arXiv · 2206.04753

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

Pavel Gumenyuk·Takahiro Hasebe·José-Luis Pérez
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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.

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Pavel Gumenyuk, Takahiro Hasebe, José-Luis Pérez. 2022-06-09. Loewner Theory for Bernstein functions I: evolution families and differential equations. https://arxiv.org/abs/2206.04753

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