Stationary local random countable sets over the Wiener noise

Vidmar, Matija (author), Warren, Jon (author)

The times of Brownian local minima, maxima and their union are three distinct examples of local, stationary, dense, random countable sets associated with classical Wiener noise. Being local means, roughly, determined by the local behavior of the sample paths of the Brownian motion, and stationary means invariant relative to the Lévy shifts of the sample paths. We answer to the affirmative Tsirelson’s question, whether or not there are any others, and develop some general theory for such sets. An extra ingredient to their structure, that of an honest indexation, leads to a splitting result that is akin to the Wiener–Hopf factorization of the Brownian motion at the minimum (or maximum) and has the latter as a special case. Sets admitting an honest indexation are moreover shown to have the property that no stopping time belongs to them with positive probability. They are also minimal: they do not have any non-empty proper local stationary subsets. Random sets, of the kind studied in this paper, honestly indexed or otherwise, give rise to nonclassical one-dimensional noises, generalizing the noise of splitting. Some properties of these noises and the inter-relations between them are investigated. In particular, subsets are connected to subnoises.

• Language: English
• Keywords: random countable sets, two-sided Brownian motion, locality, stationarity, zero-one law, thick sets, splitting, noise, spectral measure
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: Str. 1063-1129
• Numbering: Vol. 188, iss. 3/4
• PID: 20.500.12556/RUL-154963
• UDC: 519.2
• ISSN on article: 0178-8051
• DOI: 10.1007/s00440-023-01227-3
• COBISS.SI-ID: 166137603
• Publication date in RUL: 12.03.2024

Record is a part of a journal

• Title: Probability theory and related fields
• Shortened title: Probab. theory relat. fields
• Publisher: Springer Nature
• ISSN: 0178-8051
• COBISS.SI-ID: 26171392

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Projects

• Funder: Other - Other funder or multiple funders
• Funding programme: University of Warwick, Fernandes Fellowship

• Funder: ARRS - Slovenian Research Agency
• Project number: P1-0402
• Name: Matematična fizika