Stationary local random countable sets over the Wiener noise

TL;DR

This paper investigates local, stationary, dense, random countable sets related to Wiener noise, answering Tsirelson's question about their existence and developing a general theory including their structure, properties, and connection to nonclassical noises.

Contribution

It confirms the existence of such sets beyond Brownian extrema, introduces the concept of honest indexation, and explores their structural and probabilistic properties.

Findings

Existence of new local stationary random sets beyond extrema

Sets with honest indexation do not contain stopping times with positive probability

These sets lead to nonclassical one-dimensional noises

Abstract

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 L\'evy 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.