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Nonlinear resolvents and decreasing Loewner chains

Ikkei Hotta, Sebastian Schleißinger, Toshiyuki Sugawa

Abstract

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.

Nonlinear resolvents and decreasing Loewner chains

Abstract

In this article we prove that nonlinear resolvents of infinitesimal generators on bounded and convex subdomains of are decreasing Loewner chains. Furthermore, we consider the problem of the existence of nonlinear resolvents on unbounded convex domains in . 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.
Paper Structure (9 sections, 23 theorems, 59 equations, 1 figure)

This paper contains 9 sections, 23 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Semigroups and nonlinear resolvents
  3. Existence of nonlinear resolvents in unbounded domains
  4. Proof of Theorem \ref{['thm:unbdd']}
  5. Parallel strips
  6. Decreasing Loewner chains
  7. Upper half-plane and Denjoy-Wolff point at infinity
  8. Additive convolution
  9. Multiplicative convolution

Key Result

Theorem 1.1

Let $D\subset \mathbb C^n$ be a domain which is bounded and convex. Let $G$ be an infinitesimal generator on $D$. Then there exists a unique solution $(f_t)$ to The mappings $f_t$ are the nonlinear resolvents of $G$ and they form a decreasing Loewner chain.

Figures (1)

  • Figure 1: $F$-transform of the semicircle distribution $W(0,1)$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3: RS97
  • Example 2.4
  • Example 2.5
  • ...and 47 more