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

TL;DR

This paper extends the concept of stochastic noise of splitting to a broader class of domains, analyzing the stability of Brownian maxima and the conditions under which the basic noise can be extended.

Contribution

It introduces a maximal class of domains for noise extension and characterizes the stability conditions related to Brownian maxima and resampling effects.

Findings

The noise extension is possible when the domain's maxima are unaffected by resampling.

Stable domains include open sets and their measure-theoretic equivalents.

Some domains are totally unstable, preventing extension.

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 $\sigma$-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.