Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Factorizing random sets and type III Arveson systems

Remus Floricel

Abstract

We develop a representative-level framework for the Liebscher-Tsirelson random-set construction of Arveson systems from stationary factorizing measure types. We introduce the notion of a measurable factorizing family of probability measures on hyperspaces of closed subsets of time intervals and prove that every such family canonically generates an Arveson system. Within this framework we obtain a purely measure-theoretic characterization of spatiality: positive normalized units correspond exactly to dominated families of measures that factorize strictly. We then present a general mechanism for constructing type III Arveson systems via infinite products of measurable factorizing families. Starting from a type II$_0$ seed satisfying a quantitative Hellinger-smallness condition, we form a marked infinite product indexed by $[0,1]\times\mathbb N$ and show, using Kakutani's criterion, that the resulting product system admits no units. This yields a robust construction principle for type III random-set systems. As an application we analyze zero sets of Brownian motion. After anchor-adapted localization and Palm-type uniformization, the Brownian seed satisfies the required overlap estimates, and the associated infinite-product construction produces explicit examples of type III random-set systems, as anticipated in the work of Tsirelson and Liebscher.

Factorizing random sets and type III Arveson systems

Abstract

We develop a representative-level framework for the Liebscher-Tsirelson random-set construction of Arveson systems from stationary factorizing measure types. We introduce the notion of a measurable factorizing family of probability measures on hyperspaces of closed subsets of time intervals and prove that every such family canonically generates an Arveson system. Within this framework we obtain a purely measure-theoretic characterization of spatiality: positive normalized units correspond exactly to dominated families of measures that factorize strictly. We then present a general mechanism for constructing type III Arveson systems via infinite products of measurable factorizing families. Starting from a type II seed satisfying a quantitative Hellinger-smallness condition, we form a marked infinite product indexed by and show, using Kakutani's criterion, that the resulting product system admits no units. This yields a robust construction principle for type III random-set systems. As an application we analyze zero sets of Brownian motion. After anchor-adapted localization and Palm-type uniformization, the Brownian seed satisfies the required overlap estimates, and the associated infinite-product construction produces explicit examples of type III random-set systems, as anticipated in the work of Tsirelson and Liebscher.
Paper Structure (18 sections, 45 theorems, 345 equations)

This paper contains 18 sections, 45 theorems, 345 equations.

Table of Contents

  1. Introduction
  2. Main results and organization.
  3. Background on Arveson systems
  4. Notation and Conventions.
  5. Random-set systems
  6. Closed-set spaces and factorizing families of measures
  7. Relation to Liebscher's stationary factorizing measure types
  8. From factorizing families to algebraic Arveson systems
  9. Compatibility with Liebscher--Tsirelson construction
  10. Measurability: from measurable families to Arveson systems
  11. Units and Type
  12. Spatiality
  13. Fully $L$--spread Poisson random closed sets
  14. Type III random-set systems
  15. Hellinger-small seeds
  16. ...and 3 more sections

Key Result

Lemma 2.1

The collection $\{V_t(U): U\in\mathcal{U}_t^{\mathbb Q}\}$ generates $\Sigma_t$ for every $t>0$.

Theorems & Definitions (102)

  • Lemma 2.1
  • proof
  • Definition 2.3
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • proof
  • Proposition 2.10
  • ...and 92 more