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Ikkei Hotta

Publications and source records attributed to Ikkei Hotta.

At least 19 recordsLinked to original sources
arXiv2026

Boundary geometry and linear accessibility of functions with positive real derivative

We study the boundary geometry of the Noshiro--Warschawski class $\mathcal{R}$. Using the geometric structure of close-to-convex domains and their relation to Loewner chains, we investigate the boundary behavior of functions in $\mathcal{R}$. In particular, we discuss the relation between spherical length and local connectedness, and show that the boundary of the image domain of a function in $\mathcal{R}$ need not be locally connected. We also revisit the classical fact that $\mathcal{R}$ is not contained in the class $\mathcal{S}^{*}$ of starlike functions and give a simple explicit example of a function in $\mathcal{R}\setminus\mathcal{S}^*$.

math.CV↗
arXiv2026

Additive processes on the real line and Loewner chains

This paper investigates additive processes with respect to several different independences in non-commutative probability in terms of the convolution hemigroups of the distributions of the increments of the processes. In particular, we focus on the relation of monotone convolution hemigroups and Loewner chains, a special kind of family of conformal mappings, on the upper half-plane. Generalizing the celebrated Loewner differential equation, we formulate an integral equation and the concept of ``generator'' for any Loewner chain of reciprocal Cauchy transforms. This generalization enables us to remove the assumption of the absolute continuity of Loewner chains which had been imposed in the literature. The locally uniform convergence of Loewner chains is then equivalent to a suitable convergence of generators. Using generators, we define homeomorphisms between the aforementioned class of Loewner chains, the set of monotone convolution hemigroups, and the set of classical convolution hemigroups on the real line. We also discuss similar homeomorphisms to free, boolean and anti-monotone convolution hemigroups on the real line.

math.PR↗
arXiv2026

Topology and convergence on the space of measure-valued functions

In these notes, uniform convergence on compacta is studied on the space of functions taking values in the set of finite Borel measures. Related limit theorems, including Lévy's continuity theorem and functional limit theorems for (classical and non-commutative) additive processes, are also described. N.B.: the contents of this manuscript have been incorporated into another manuscript (arXiv:2412.18742).

math.PR↗
arXiv2024

Nonlinear resolvents and decreasing Loewner chains

In this article we prove that nonlinear resolvents of infinitesimal generators on bounded and convex subdomains of $\C^n$ are decreasing Loewner chains. Furthermore, we consider the problem of the existence of nonlinear resolvents on unbounded convex domains in $\C$. In the case of the upper half-plane, we obtain a complete solution by using that nonlinear resolvents of certain generators correspond to semigroups of probability measures with respect to free convolution.

math.CV↗
arXiv2022

Continuous evolution families

Recently in relation to the theory of non-commutative probability, a notion of evolution families $\{ω_{s,t}\}_{s \le t}$ is generalized that are only continuous in parameters, namely $(s,t) \mapsto ω_{s,t}$ is continuous with respect to locally uniform convergence on a planar domain. In this article we present various equivalence conditions to the continuous evolution families concerned with the left and right parameters. We also provide an example of a discontinuous evolution family in the last section.

math.CV↗
arXiv2022

On freely quasi-infinitely divisible distributions

Inspired by the notion of quasi-infinite divisibility (QID), we introduce and study the class of freely quasi-infinitely divisible (FQID) distributions on $\mathbb{R}$, i.e. distributions which admit the free Lévy-Khintchine-type representation with signed Lévy measure. We prove several properties of the FQID class, some of them in contrast to those of the QID class. For example, a FQID distribution may have negative Gaussian part, and the total mass of its signed Lévy measure may be negative. Finally, we extend the Bercovici-Pata bijection, providing a characteristic triplet, with the Lévy measure having nonzero negative part, which is at the same time classical and free characteristic triplet.

math.PR↗
arXiv2021

Additive processes on the unit circle and Loewner chains

This paper defines the notion of generators for a class of decreasing radial Loewner chains which are only continuous with respect to time. For this purpose, "Loewner's integral equation" which generalizes Loewner's differential equation is defined and analyzed. The definition of generators is motivated by the Lévy-Khintchine representation for additive processes on the unit circle. Actually, we can and do introduce a homeomorphism between the above class of Loewner chains and the set of the distributions of increments of additive processes equipped with suitable topologies. On the other hand, from the viewpoint of non-commutative probability theory, the above generators also induce bijections with some other objects: in particular, monotone convolution hemigroups and free convolution hemigroups. Finally, the generators of Loewner chains constructed from free convolution hemigroups via subordination are computed.

math.CV↗
arXiv2020

Limits of radial multiple SLE and a Burgers-Loewner differential equation

We consider multiple radial SLE as the number of curves tends to infinity. We give conditions that imply the tightness of the associated processes given by the Loewner equation. In the case of equal weights, the infinite-slit limit is described by a Loewner equation whose Herglotz vector field is given by a Burgers differential equation. Furthermore, we investigate a more general form of the Burgers equation. On the one hand, it appears in connection with semigroups of probability measures on the unit circle with respect to free convolution. On the other hand, the Burgers equation itself is also a Loewner differential equation for certain subordination chains.

math.PR↗
arXiv2019

Univalent functions with quasiconformal extensions: Becker's class and estimates of the third coefficient

We investigate univalent functions $f(z)=z+a_2z^2+a_3z^3+\ldots$ in the unit disk $\mathbb D$ extendible to $k$-q.c.(=quasiconformal) automorphisms of $\mathbb C$. In particular, we answer a question on estimation of $|a_3|$ raised by Kühnau and Niske [Math. Nachr. 78 (1977) 185-192]. This is one of the results we obtain studying univalent functions that admit q.c.-extensions via a construction, based on Loewner's parametric representation method, due to Becker [J. Reine Angew. Math. 255 (1972) 23-43]. Another problem we consider is to find the maximal $k_*\in(0,1]$ such that every univalent function $f$ in $\mathbb D$ having a $k$-q.c. extension to $\mathbb C$ with $k\leqslant k_*$ admits also a Becker q.c.-extension, possibly with a larger upper bound for the dilatation. We prove that $k_*>1/6$. Moreover, we show that in some cases, Becker's extension turns out to be the optimal one. Namely, given any $k\in(0,1)$, to each finite Blaschke product there corresponds a univalent function $f$ in $\mathbb D$ that admits a Becker $k$-q.c. extension but no $k'$-q.c. extensions to $\mathbb C$ with $k'<k$.

math.CV↗
arXiv2019

Loewner chains with quasiconformal extensions: an approximation approach

A new approach in Loewner Theory proposed by Bracci, Contreras, Díaz-Madrigal and Gumenyuk provides a unified treatment of the radial and the chordal versions of the Loewner equations. In this framework, a generalized Loewner chain satisfies the differential equation $$ \partial_{t}f_{t}(z) = (z - τ(t))(1-\overline{τ(t)}z)\partial_{z}f_{t}(z)p(z,t), $$ where $τ: [0,\infty) \to \overline{\mathbb{D}}$ is measurable and $p$ is called a Herglotz function. In this paper, we will show that if there exists a $k \in [0,1)$ such that $p$ satisfies $$ |p(z,t) - 1| \leq k |p(z,t) + 1| $$ for all $z \in \mathbb{D}$ and almost all $t \in [0,\infty)$, then $f_{t}$ has a $k$-quasiconformal extension to the whole Riemann sphere for all $t \in [0,\infty)$. The radial case ($τ=0$) and the chordal case ($τ=1$) have been proven by Becker [J. Reine Angew. Math. \textbf{255} (1972), 23-43] and Gumenyuk and the author (Math. Z. \textbf{285} (2017), no.3, 1063--1089). In our theorem, no superfluous assumption is imposed on $τ\in \overline{\mathbb{D}}$. As a key foundation of our proof is an approximation method using the continuous dependence of evolution families.

math.CV↗
arXiv2018

Problems related to conformal slit-mappings

In this note we discuss some problems related to conformal slit-mappings. On the one hand, classical Loewner theory leads us to questions concerning the embedding of univalent functions into slit-like Loewner chains. On the other hand, a recent result from monotone probability theory motivates the study of univalent functions from a probabilistic perspective.

math.CV↗
arXiv2018

Hydrodynamic Limit of Multiple SLE

Recently del Monaco and Schleißinger addressed an interesting problem whether one can take the limit of multiple Schramm--Loewner evolution (SLE) as the number of slits $N$ goes to infinity. When the $N$ slits grow from points on the real line ${\mathbb{R}}$ in a simultaneous way and go to infinity within the upper half plane ${\mathbb{H}}$, an ordinary differential equation describing time evolution of the conformal map $g_t(z)$ was derived in the $N \to \infty$ limit, which is coupled with a complex Burgers equation in the inviscid limit. It is well known that the complex Burgers equation governs the hydrodynamic limit of the Dyson model defined on ${\mathbb{R}}$ studied in random matrix theory, and when all particles start from the origin, the solution of this Burgers equation is given by the Stieltjes transformation of the measure which follows a time-dependent version of Wigner's semicircle law. In the present paper, first we study the hydrodynamic limit of the multiple SLE in the case that all slits start from the origin. We show that the time-dependent version of Wigner's semicircle law determines the time evolution of the SLE hull, $K_t \subset {\mathbb{H}} \cup {\mathbb{R}}$ , in this hydrodynamic limit. Next we consider the situation such that a half number of the slits start from $a>0$ and another half of slits start from $-a < 0$, and determine the multiple SLE in the hydrodynamic limit. After reporting these exact solutions, we will discuss the universal long-term behavior of the multiple SLE and its hull $K_t$ in the hydrodynamic limit.

math-ph↗
arXiv2016

Tightness results for infinite-slit limits of the chordal Loewner equation

In this note we consider a multi-slit Loewner equation with constant coefficients that describes the growth of multiple SLE curves connecting $N$ points on $\mathbb{R}$ to infinity within the upper half-plane. For every $N\in\mathbb{N}$, this equation provides a measure valued process $t\mapsto \{α_{N,t}\},$ and we are interested in the limit behaviour as $N\to\infty.$ We prove tightness of the sequence $\{α_{N,t}\}_{N\in\mathbb{N}}$ under certain assumptions and address some further problems.

math.CV↗
arXiv2015

Chordal Loewner chains with quasiconformal extensions

In 1972, Becker [J. Reine Angew. Math. 255 (1972), 23-43] discovered a construction of quasiconformal extensions making use of the classical radial Loewner chains. In this paper we develop a chordal analogue of Becker's construction. As an application, we establish new sufficient conditions for quasiconformal extendibility of holomorphic functions and give a simplified proof of one well-known result by Becker and Pommerenke for functions in the half-plane [J. Reine Angew. Math. 354 (1984), 74-94].

math.CV↗
arXiv2015

Quasiconformal extendibility of integral transforms of Noshiro-Warschawski functions

Since the nonlinear integral transforms $J_α[f](z) = \int_{0}^{z}(f'(u))^α du$ and $I_α[f](z) =\int_0^z (f(u)/u)^α du$ with a complex number $α$ have been introduced, a great number of studies were dedicated to deriving sufficient conditions for univalence on the unit disk. On the other hand, little is known about the conditions that $J_α[f]$ or $I_α[f]$ produces a holomorphic univalent function in the unit disk which extends to a quasiconformal map on the complex plane. In this paper we discuss quasiconformal extendibility of the integral transforms $J_α[f]$ and $I_α[f]$ for holomorphic functions which satisfy the Noshiro-Warschawski criterion. Various approaches using pre-Schwarzian derivatives, differential subordinations and Loewner theory are taken to this problem.

math.CV↗
arXiv2014

Locally one-to-one harmonic functions with starlike analytic part

Let $L_H$ denote the set of all normalized locally one-to-one and sense-preserving harmonic functions in the unit disc $Δ$. It is well-known that every complex-valued harmonic function in the unit disc $Δ$ can be uniquely represented as $f = h + \overline{g}$, where $h$ and $g$ are analytic in $Δ$. In particular the decomposition formula holds true for functions of the class $L_H$. For a fixed analytic function $h$, an interesting problem arises - to describe all functions $g$, such that $f$ belongs to $L_H$. The case when $f\in L_H$ and $h$ is the identity mapping was considered [2]. More general results are given in [3], where $f\in L_H$ and letting $h$ to be a convex analytic mapping. The focus of our present research is to characterize the set of all functions $f\in L_H$ having starlike analytic part $h$. In this paper, we provide coefficient, distortion and growth estimates of $g$. We also give growth and Jacobian estimates of $f$.

math.CV↗
arXiv2011

Ahlfors's quasiconformal extension condition and $Φ$-likeness

The notion of $Φ$-like functions is known to be a necessary and sufficient condition for univalence. By applying the idea, we derive several necessary conditions and sufficient conditions for that an analytic function defined on the unit disk is not only univalent but also has a quasiconformal extension to the Riemann sphere, as generalizations of well-known univalence and quasiconformal extension criteria, in particular, Ahlfors's quasiconformal extension condition.

math.CV↗