论文标题
伯恩斯坦功能的Loewner理论I：进化家庭和微分方程
Loewner Theory for Bernstein functions I: evolution families and differential equations
论文作者
论文摘要
全体形态功能的单参数半群自然出现在复杂分析的各种应用中，尤其是在（时间上）同质的马尔可夫过程中。在不均匀环境中的单参数半群的合适类似物是（反向）进化家族的概念。在本文中，我们研究了伯恩斯坦功能形成的进化族，该家族起着拉普拉斯指数在不均匀连续分支过程中的作用。特别是，我们表征了所有产生此类进化族的Herglotz载体场，并提供了与Silverstein的代表公式相当的定性描述，用于伯恩斯坦功能的单参数半群的无限发电机。我们还为全态自图家族建立了几个足够的条件，满足了进化家族定义的代数部分，绝对是连续的，因此被描述为通用loewner -kufarev差异方程的解决方案。然后将这些结果大多数应用于续集[https://doi.org/10.48550/arxiv.2211.12442]中，以研究连续态分支过程。
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.
