Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extending the noise of splitting to its completion and stability of Brownian maxima

Matija Vidmar, Jon Warren

TL;DR

This work advances the theory of stochastic noises by constructing and characterizing the largest extension of the one-dimensional splitting noise to a completion of its range, thus answering Tsirelson's program for item (b) in this nonclassical setting. The authors develop a spectral, Boolean-algebra framework in which extendable domains are exactly the max-enumerable sets, tying extendability to stability properties of Brownian maxima under resampling outside the domain. They show the extension exists and is nontrivial, giving explicit descriptions of the extended σ-fields on max-enumerable domains and proving several structural results, including that the extension is maximal within the closure of the original noise and respects the stable part. They also reveal a rich stability-sensitivity landscape: some domains yield max-total stability while others yield max-total instability, with density-based criteria and subordinated-process constructions providing concrete examples of both. The results have implications for understanding nonclassical noises and offer tools for extending other noises, illustrating deep connections between Brownian path properties and noise completion.

Abstract

The stochastic noise of splitting, defined initially on the (basic) algebra of finite unions of intervals of the real line, is extended to a largest class of domains. The $σ$-fields of this largest extension constitute the completion, in the sense of noise-type Boolean algebras, of the range of the unextended (basic) noise. The basic noise extends to a given measurable domain precisely when a certain stability property is met: the times at which a Brownian motion has local maxima which fall inside the domain must remain unaffected under resampling of the Brownian increments outside the domain; together with the same being true for the complement of the domain. A set that is equal to an open set modulo a Lebesgue negligible one, with the same holding of its complement, has this stability property, but others have it too: the extension is non-trivial. Some domains are totally unstable with respect to the indicated resampling, and to them the extension cannot be made.

Extending the noise of splitting to its completion and stability of Brownian maxima

TL;DR

This work advances the theory of stochastic noises by constructing and characterizing the largest extension of the one-dimensional splitting noise to a completion of its range, thus answering Tsirelson's program for item (b) in this nonclassical setting. The authors develop a spectral, Boolean-algebra framework in which extendable domains are exactly the max-enumerable sets, tying extendability to stability properties of Brownian maxima under resampling outside the domain. They show the extension exists and is nontrivial, giving explicit descriptions of the extended σ-fields on max-enumerable domains and proving several structural results, including that the extension is maximal within the closure of the original noise and respects the stable part. They also reveal a rich stability-sensitivity landscape: some domains yield max-total stability while others yield max-total instability, with density-based criteria and subordinated-process constructions providing concrete examples of both. The results have implications for understanding nonclassical noises and offer tools for extending other noises, illustrating deep connections between Brownian path properties and noise completion.

Abstract

The stochastic noise of splitting, defined initially on the (basic) algebra of finite unions of intervals of the real line, is extended to a largest class of domains. The -fields of this largest extension constitute the completion, in the sense of noise-type Boolean algebras, of the range of the unextended (basic) noise. The basic noise extends to a given measurable domain precisely when a certain stability property is met: the times at which a Brownian motion has local maxima which fall inside the domain must remain unaffected under resampling of the Brownian increments outside the domain; together with the same being true for the complement of the domain. A set that is equal to an open set modulo a Lebesgue negligible one, with the same holding of its complement, has this stability property, but others have it too: the extension is non-trivial. Some domains are totally unstable with respect to the indicated resampling, and to them the extension cannot be made.
Paper Structure (23 sections, 31 theorems, 187 equations)

This paper contains 23 sections, 31 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Motivation and agenda
  3. The landscape in broad strokes
  4. Continuous products and noises
  5. Stability and sensitivity
  6. Boolean algebras of $\sigma$-fields, closure and completion
  7. Tanaka's SDE and the splitting noise
  8. Highlights and discussion of results
  9. Article structure
  10. Preliminaries
  11. Miscellaneous general notation, vocabulary, conventions
  12. Lattice of sub-$\sigma$-fields and noise(-type) Boolean algebras
  13. The noise of splitting
  14. Spectrum of a noise
  15. Spectral structure of the splitting noise
  16. ...and 8 more sections

Key Result

Proposition 3.1

Let $A\subset \mathbb{R}$ be an $\mathscr{L}$-measurable set. Then $S_A$ is $\mu$-measurable. Furthermore, $S_{\mathbb{R}\backslash A}$ is $\mu$-conegligible [i.e. $\mu(S\backslash S_{\mathbb{R}\backslash A})=0$; we will say that "$A$ is negligible for the noise"] iff $A$ is $\mathscr{L}$-negligible

Theorems & Definitions (79)

  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof : Proof of Proposition \ref{['proposition:meet-and-join']}
  • ...and 69 more