arXiv subjects
Publications and source records attributed to Sebastian Schleißinger.
In this note we regard non-commutative probability theory with operator-valued expectation. We show that the moment generating functions of distributions coming from monotone increment processes of unitary random variables yield biholomorphic mappings on certain higher dimensional unit balls.
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.
Quantum probability theory and complex analysis for children.
We prove one-to-one correspondences between certain decreasing Loewner chains in the upper half-plane, a special class of real-valued Markov processes, and quantum stochastic processes with monotonically independent additive increments. This leads us to a detailed investigation of probability measures on $\mathbb{R}$ with univalent Cauchy transform. We discuss several subclasses of such measures and obtain characterizations in terms of analytic and geometric properties of the corresponding Cauchy transforms. Furthermore, we obtain analogous results for the setting of decreasing Loewner chains in the unit disk, which correspond to quantum stochastic processes of unitary operators with monotonically independent multiplicative increments.
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.
We regard a certain type of Loewner's differential equation from a quantum probability point of view and approximate the underlying quantum process by the adjacency matrices of growing graphs which arise from the comb product of certain spidernets.
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.
In [5], O. Bauer interpreted the chordal Loewner equation in terms of non-commutative probability theory. We follow this perspective and identify the chordal Loewner equations as the non-autonomous versions of evolution equations for semigroups in monotone and anti-monotone probability theory. We also look at the corresponding equation for free probability theory.
We consider support points of the class $S^0(\mathbb{D}^n)$ of normalized univalent mappings on the polydisc $\mathbb{D}^n$ with parametric representation and we prove sharp estimates for coefficients of degree 2.
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.
Let $\mathbb D$ be the unit disc and $z_0\in\mathbb D.$ We determine the value range $\{f(z_0)\,|\, f\in \mathcal{R}^\geq\}$, where $\mathcal{R}^\geq$ is the set of holomorphic functions $f:\mathbb D\to\mathbb D$ with $f(0)=0$ and $f'(0)\geq0$ that have only real coefficients in their power series expansion around $0$, and the smaller set $\{f(z_0)\,|\, f\in \mathcal{R}^\geq, \text{$f$ is typically real}\}.$ Furthermore, we describe a third value range $\{ f(z_0) \,|\, f \in \mathcal{I}\}$, where $\mathcal{I}$ consists of all univalent self-mappings of the upper half-plane $\mathbb{H}$ with hydrodynamical normalization which are symmetric with respect to the imaginary axis.
Let $γ_1,γ_2:[0,T]\to \overline{\mathbb{D}}\setminus\{0\}$ be parametrizations of two slits $Γ_1:=γ(0,T], Γ_2=γ_2(0,T]$ such that $Γ_1$ and $Γ_2$ are disjoint. \\ Let $g_t$ to be the unique normalized conformal mapping from $\mathbb{D}\setminus (γ_1[0,t]\cup γ_2[0,t])$ onto $\mathbb{D}$ with $g_t(0)=0,$ $g'_t(0)>0$. Furthermore, for $k=1,2$, denote by $h_{k;t}$ the unique normalized conformal mapping from $\mathbb{D}\setminus γ_k[0,t]$ onto $\mathbb{D}$ with $h_{k;t}(0)=0,$ ${h'_{k;t}(0)}>0$.\\ Loewner's famous theorem (\cite{Loewner:1923}) can be stated in the following way: The function $t\mapsto h_{k;t}$ is differentiable at $t_0$ if and only if $t\mapsto \log(h_{k;t}'(0))$ is differentiable at $t_0$.\\ In this paper we compare the differentiability of $t\mapsto h_{k;t}$ with that of $t\mapsto g_t.$ We show that the situation is more complicated in the case $t_0=0$ with $γ_1(0)=γ_2(0).$\\ Furthermore, we also look at this problem in the case of a multiply connected domain with its corresponding Komatu-Loewner equation.
We consider the chordal Loewner differential equation for multiple slits in the upper half-plane and relations between the pointwise Hölder continuity of the driving functions and the generated hulls. The first result generalizes a result of Lind that gives a sufficient condition for driving functions to generate simple curves. The second result translates the property that the hulls locally look like straight lines at their starting points into a condition for the driving functions.
We prove that any disjoint union of finitely many simple curves in the upper half-plane can be generated in a unique way by the chordal multiple-slit Loewner equation with constant weights.
The radial Komatu-Loewner equation is a differential equation for certain normalized conformal mappings that can be used to describe the growth of slits within multiply connected domains. We show that it is possible to choose constant coefficients in this equation in order to generate given disjoint slits and that those coefficients are uniquely determined under a suitable normalization of the differential equation.
We describe the region $\mathcal{V}(z_0)$ of values of $f(z_0)$ for all normalized bounded univalent functions $f$ in the unit disk $\mathbb{D}$ at a fixed point $z_0 \in \mathbb{D}$. The proof is based on identifying $\mathcal{V}(z_0)$ as the reachable set of the radial Loewner differential equation. We also prove an analogous result for the upper half-plane using the chordal Loewner equation.