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The main purpose of this paper is to study time-inhomogeneous one-dimensional branching processes (mainly on a continuous but also on a discrete state space) with the help of recent achievements in Loewner Theory dealing with evolution families of holomorphic self-maps in simply connected domains of the complex plane. Under a suitable stochastic continuity condition, we show that the families of the Laplace exponents of branching processes on $[0,\infty]$ can be characterized as topological (i.e. depending continuously on the time parameters) reverse evolution families whose elements are Bernstein functions. For the case of a stronger regularity w.r.t. time, we establish a Loewner - Kufarev type ODE for the Laplace exponents and characterize branching processes with finite mean in terms of the vector field driving this ODE. Similar results are obtained for families of probability generating functions for branching processes on the discrete state space $\{0,1,2,\ldots\}\cup\{\infty\}$. In addition, we find necessary and sufficient conditions for "spatial" embeddability of such branching processes into branching processes on $[0,\infty]$. Finally, we give some probabilistic interpretations of the Denjoy - Wolff point at $0$ and at $\infty$., Reference [16] has been updated

A classical result in the theory of Loewner's parametric representation states that the semigroup $\mathfrak U_*$ of all conformal self-maps $\phi$ of the unit disk $\mathbb{D}$ normalized by $\phi(0) = 0$ and $\phi'(0) > 0$ can be obtained as the reachable set of the Loewner - Kufarev control system $$ \frac{\mathrm{d} w_t}{\mathrm{d} t}=G_t\circ w_t,\quad t\geqslant0,\qquad w_0=\mathsf{id}_{\mathbb{D}}, $$ where the control functions $t\mapsto G_t\in\mathsf{Hol}(\mathbb{D},\mathbb{C})$ form a certain convex cone. Here we extend this result to the semigroup $\mathfrak U[F]$ consisting of all conformal $\phi:\mathbb{D}\to\mathbb{D}$ whose set of boundary regular fixed points contains a given finite set $F\subset\partial\mathbb{D}$ and to its subsemigroup $\mathfrak U_\tau[F]$ formed by $\mathsf{id}_{\mathbb{D}}$ and all $\phi\in\mathfrak U[F]\setminus\{\mathsf{id}_{\mathbb{D}}\}$ with the prescribed boundary Denjoy - Wolff point $\tau\in\partial\mathbb{D}\setminus F$. This completes the study launched in [P. Gumenyuk, Preprint 2016, ArXiv:1603.04043], where the case of interior Denjoy - Wolff point $\tau\in\mathbb{D}$ was considered.

We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps in the unit disc. We introduce "regular" fractional singularities and characterize them in terms of the behavior of the associated semigroups and Koenigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the Koenigs function for having such singularities. In order to do this, we study contact points of semigroups and prove that any contact (not fixed) point of a one-parameter semigroup corresponds to a maximal arc on the boundary to which the associated infinitesimal generator extends holomorphically as a vector field tangent to this arc., Comment: A reference to [Elin, Mark; Khavinson, Dmitry; Reich, Simeon; Shoikhet, David. Linearization models for parabolic dynamical systems via Abel's functional equation. Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 2, 439--472] is added

It has been an open problem for about ten years whether every trajectory of a parabolic one-parameter semigroup in the unit disk tends to the Denjoy-Wolff point with a definite (and common for all trajectories) slope. In this paper, we give the negative answer to this question.