Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Additive processes on the real line and Loewner chains

Takahiro Hasebe, Ikkei Hotta, Takuya Murayama

TL;DR

This work builds a unified Loewner-theoretic framework for additive processes across classical, monotone, free, and boolean convolutions. By introducing generators via chordal Loewner chains and Loewner integro-differential equations, it establishes bijections between convolution hemigroups with finite second moments and between their monotone and classical counterparts, with continuity under locally uniform convergence. The theory leverages reduced generating families to translate moment evolution into generator dynamics, enabling a precise link between probabilistic limit theorems and complex-analytic evolution. The results generalize Lévy–Khintchine representations to the monotone setting, extend bijections to free/boolean cases, and connect these algebraic structures to geometric function theory through capacity and area considerations, with implications for limit theorems and SLE-inspired processes.

Abstract

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 chordal Loewner chains, a special kind of family of conformal mappings. Generalizing the celebrated Loewner differential equation, we introduce the concept of ``generator'' for a class of Loewner chains. 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 chordal Loewner chains, the set of monotone convolution hemigroups with finite second moment, and the set of classical convolution hemigroups on the real line with finite second moment. Moreover, we define similar homeomorphisms between classical, free, and boolean convolution hemigroups on the real line, but without any assumptions on the moments.

Additive processes on the real line and Loewner chains

TL;DR

This work builds a unified Loewner-theoretic framework for additive processes across classical, monotone, free, and boolean convolutions. By introducing generators via chordal Loewner chains and Loewner integro-differential equations, it establishes bijections between convolution hemigroups with finite second moments and between their monotone and classical counterparts, with continuity under locally uniform convergence. The theory leverages reduced generating families to translate moment evolution into generator dynamics, enabling a precise link between probabilistic limit theorems and complex-analytic evolution. The results generalize Lévy–Khintchine representations to the monotone setting, extend bijections to free/boolean cases, and connect these algebraic structures to geometric function theory through capacity and area considerations, with implications for limit theorems and SLE-inspired processes.

Abstract

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 chordal Loewner chains, a special kind of family of conformal mappings. Generalizing the celebrated Loewner differential equation, we introduce the concept of ``generator'' for a class of Loewner chains. 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 chordal Loewner chains, the set of monotone convolution hemigroups with finite second moment, and the set of classical convolution hemigroups on the real line with finite second moment. Moreover, we define similar homeomorphisms between classical, free, and boolean convolution hemigroups on the real line, but without any assumptions on the moments.

Paper Structure

This paper contains 49 sections, 80 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Convolutions of probability measures, convolution hemigroups and additive processes
  4. Objective of this paper
  5. Core idea
  6. Overview of main results
  7. Loewner chains and monotone CHs
  8. Loewner integro-differential equations
  9. Generators of Loewner chains and bijections between Loewner chains, classical and monotone CHs
  10. Continuity of the bijection
  11. Outline of the subsequent sections
  12. Classical convolution hemigroups
  13. A generalization of Lévy's continuity theorem
  14. Convergence of classical additive processes
  15. Reduced generating families
  16. ...and 34 more sections

Key Result

Proposition 1.4

Let $\mathfrak{c}$ be a $\ast$-infinitely divisible distribution on $\mathbb{R}$. Then there exists a unique pair $(\gamma,\eta) \in \mathbb{R} \times \mathbf{M}(\mathbb R)$ such that Conversely, for every pair $(\gamma,\eta) \in \mathbb{R} \times \mathbf{M}(\mathbb R)$ there exists a unique $\ast$-infinitely divisible distribution $\mathfrak{c}$ for which eq:CLK0_intro holds.

Theorems & Definitions (156)

  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4: Lévy-Khintchine representation
  • Proposition 1.5
  • Proposition 1.6
  • Remark 1.8
  • Theorem 1.9: Theorem \ref{['th:capacity_conti_prime']} below
  • Remark 1.10
  • Theorem 1.11: Theorems \ref{['th:DLC_to_LIE']} and \ref{['th:LIE_to_DLC']} below
  • Remark 1.12
  • ...and 146 more